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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 314)
by knarf＠cct.lsu.edu 13 Mar '12

13 Mar '12

User: knarf Date: 2012/03/13 03:36 PM Modified: / ET.tex, iop_revised.tar.gz Log: submitted version File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +2 -93 =================================================================== --- ET.tex 2012-03-13 20:22:18 UTC (rev 313) +++ ET.tex 2012-03-13 20:36:03 UTC (rev 314) @@ -121,19 +121,6 @@ \address{$^{10}$ National Science Foundation, USA} \ead{knarf(a)cct.lsu.edu} -%\thanks{{}$^{(1)}$ Department of Computer Science, -% Louisiana State University, Baton Rouge, LA 70803, USA} -%\thanks{{}$^{(2)}$ Department of Physics \& Astronomy, -% Louisiana State University, Baton Rouge, LA 70803, USA} -%\thanks{{}$^{(3)}$ Center for Computation \& Technology, -% Louisiana State University, Baton Rouge, LA 70803, USA} -%\thanks{{}$^{(4)}$ Center for Computational Relativity \& Gravitation, -% School of Mathematical Sciences, -% Rochester Institute of Technology, Rochester, New York 14623, USA} - -% TODO: Already add a few affiliations which we know are going -% to be used - \begin{abstract} We describe the Einstein Toolkit, a community-driven, freely accessible computational infrastructure intended for use in numerical relativity, @@ -156,9 +143,6 @@ \pacs{04.25.D-, 04.30.-w, 04.70.-s, 07.05.Tp, 95.75.Pq} \maketitle -% We will remove this at the end, but we leave it in for ourselves now -%\tableofcontents - \section{Introduction} Scientific progress in the field of numerical relativity has always @@ -625,8 +609,6 @@ The Simulation Factory supports and simplifies three kinds of operations: -%RH: any of the itemize, decription, enumerate environments indents its body -%by the same amount as \parindent \begin{description} \item[1. Remote Access.] The actual access commands and authentication methods differ between systems, as do the user names that a person @@ -1562,10 +1544,6 @@ $T^{\mu \nu}$ is the stress-energy tensor, defined in~\eref{eq:Tmunu} as $T^{\mu \nu} = \rho h u^{\,\mu} u^{\,\nu} + P g^{\,\mu \nu}$, where $u^{\,\mu}$ is the four-velocity and $\rho$ the rest-mass density. -%T^{\mu \nu} = \rho h u^{\,\mu} u^{\,\nu} + P g^{\,\mu \nu} $ is the -%stress-energy tensor. The quantity $ h = 1 + \epsilon + P / \rho $ is -%the specific enthalpy, $P$ is the fluid pressure and $\epsilon$ is the -%specific internal energy. The 3-velocity $v^i$ may be calculated in the form \begin{equation} @@ -1793,7 +1771,6 @@ \subsection{Equations of State} \label{sec:eoss} -%%% Written by Christian Ott An equation of state connecting the primitive state variables is needed to close the system of GR hydrodynamics equations. The module @@ -1807,8 +1784,6 @@ P = K\rho^\Gamma\,\,, \end{equation} where $K$ is the polytropic constant -% RH: poly.const. is correct term according to arXiv:gr-qc/0211028 below Eq. -% (9.4) and $\Gamma$ is the adiabatic index, is appropriate for adiabatic (= isentropic) evolution without shocks. When using the polytropic EOS, one does not need to evolve the @@ -1912,7 +1887,6 @@ hypersurface. The module \codename{EHFinder} is also available to search an evolved spacetime for the globally defined event horizons. -% Event horizon \codename{EHFinder}~\cite{Diener:2003jc} evolves a null surface backwards in time given an initial guess (e.g., the last apparent horizon) which will, in the vicinity of an event @@ -2339,7 +2313,7 @@ point $x''$ for which there is actual data stored. In this example, two reflection symmetries along the horizontal and vertical axis are present. Notice how the vector components change in - transformations $A$ and $B$.} % Image courtesy of Steve White.} + transformations $A$ and $B$.} % Image courtesy of Steve White. \label{fig:faces} \end{figure} @@ -2415,12 +2389,6 @@ \begin{center} \includegraphics[width=0.4\textwidth]{rot180-grid} \end{center} - % ES: This discussion also mentioned buffer points, but was the - % only place we mention buffer points. Also, the discussion below - % is quite complex even without buffer points, so I don't mention - % them any more. (This is technically correct, because the - % algorithm described below doesn't know about buffer points -- - % these are handled before and afterwards.) \caption{Example of a grid layout created by \codename{CarpetRegrid2}. This figure shows two refinement levels, a coarse (big red circles) and a fine one (small black @@ -2666,7 +2634,7 @@ \caption{Initial data parameters and initial ADM mass for a non-spinning equal mass BH binary system. The punctures are located on the $x$-axis at positions $x_1$ and $x_2$, with puncture bare -mass parameters $m_1 = m_2 = m$, and momenta %$\pm\vec p$. +mass parameters $m_1 = m_2 = m$, and momenta $\vec p_1 = -\vec p_2 = \vec p$. } \end{table} @@ -2718,12 +2686,6 @@ resolution of $2.6 \times 10^{-2}$ radians, and an error of the same order, since the surface integrals were calculated by midpoint rule -- a first order accurate method. -%BCM: This is certainly not correct. Give a better -% thought later: -%Note however that this error is still -%%at least an order of magnitude -%smaller than, for example, the metric truncation error -%at this extraction region ($\sim 6.25 \times 10^{-2}$). In order to evaluate the convergence of the numerical solution, we ran five simulations with different @@ -2742,34 +2704,11 @@ potentially introduce, but did not observe any noticeable difference and we report here on results from the higher-order option. -%JF: Do we need this at all, if we don't show a traditional convergence plot? -%\BCM{Move this elsewhere too -- starts here} -%Since the grid spacings -%does not follow a $2:1$ ratio, the convergence factor has -%a different functional form than the usual $2^p$ for a $p$-th accurate -%finite difference scheme: -% -%\begin{equation} -%Q = \frac{b^p - a^p}{a^p - 1}, -%\end{equation} -%where $h_{\rm high}=h$, $h_{\rm med}=a\times h$ and -%$h_{\rm low} = b \times h$. In our case, $a$ and $b$ -%are $1.25$ and $1.5$, respectively. A $4$-th order convergent -%scheme for our grid spacings would give then a convergence -%factor of $Q = 1.818$, while a $6$-th and $8$-th order one, -%$2.691$ and $3.965$, respectively. -%\BCM{stops here. Remember to amend the text appropriately} - In Figure~\ref{fig:amp_phs_convergence}, we show the convergence of the amplitude and phase of the Weyl scalar by plotting the logarithm of the absolute value of the differences between two levels of resolution. The differences clearly converge to zero as the resolution is increased. -%BCM: Until we have clearer argument on why we have superconvergence, -% I am commenting out the following sentence: -%We also show that the most appropriate convergence factor -%seems to be the one corresponding to an $8$-th order accurate finite -%difference approximation. We also indicate on both plots the time at which the gravitational wave frequency reaches $\omega=0.2/\mathrm{M}$. This reference time has been suggested, for example in the NRAR\cite{NRAR:web} @@ -3012,8 +2951,6 @@ with $\approx$ 60, 80, 120, and 240 grid points. The units of time on the upper and lower $x$-axes are identical to those of Figure~\ref{fig:tov_collapse_radii}. - % initial star radius is: 6M, resolutions are 0.2,0.15,0.1,0.05 on the box - % (radius 8M) covering the star } \end{figure} In Figure~\ref{fig:tov_collapse_radii}, we plot the approximate @@ -3138,17 +3075,6 @@ approximate ways (see, e.g.,~\cite{Ott:2006eu,Farris:2008fe,Sekiguchi:2011zd,Sekiguchi:2011mc}). -%%% One of the desirable additions of physics is a proper treatment of -%%% radiation, in particular neutrinos. Another desirable addition would be some -%%% approximation of emission of electro-magnetic waves. Radiation transport is, -%%% even compared to the full GRMHD problem, computationally very expensive, -%%% especially in three dimensions. - -%%% CDO: It is not appropriate to mention a grant like PetaCactus as an ``NSF Project''. -%%% Please leave this out. -%%% Several members of the CIGR team are involved -%%% in the NSF project PetaCactus, which aims to explore these possibilities. - Besides the addition of new physics modules, existing techniques require improvement. One example is the need for the gauge invariant extraction of gravitational waves from simulation spacetimes as realized @@ -3183,19 +3109,6 @@ (e.g.,~\cite{Tchekhovskoy:2007zn}), but still await implementation in fully dynamical simulations. -%%% CDO: We have not even talked about the current MHD implementation, -%%% how can we talk about its improvement? -%%% -%%% A second example for an improvement of an existing method is extending the -%%% current implementation of ideal magnetohydrodynamics to the case of non-zero -%%% resistivity. Such implementations exist -%%% and are described in the literature~\cite{???} \todo{cite}, but none have been provided freely -%%% within the Einstein Toolkit until the recently released ``Maxwell'' version. -%%% Resistive MHD is important for a number of astrophysical scenarios, one of -%%% which is the merger of two magnetized NSs, a candidate for short gamma-ray -%%% bursts. -%%% \todo{reword - make clear that we describe Curie} - Yet another important goal is to increase the scalability of the {\tt Carpet} AMR infrastructure. As we have shown, good scaling is limited to only a few thousand processes for some of the most widely used @@ -3245,11 +3158,7 @@ authors and do not necessarily reflect the views of the National Science Foundation. -% Take this out when using natbib -%\section*{References} - \bibliographystyle{iopart-num-edit} -%\bibliographystyle{amsplain-url} % all references should be in manifest/einsteintoolkit \bibliography{manifest/einsteintoolkit} File [modified]: iop_revised.tar.gz Delta lines: +0 -0 =================================================================== (Binary files differ)

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[svn:einsteintoolkit] Workshop_Spring_2012/ (Rev. 4)
by tanja.bode＠physics.gatech.edu 13 Mar '12

13 Mar '12

User: tbode Date: 2012/03/13 02:19 PM Added: /flyer/ etk-widescreen.png Log: Added Dennis' widescreen for slideshow-based advertisement. Directory Changes: Directory: /svn:mime-type/ ========================== + application/octet-stream File Changes: Directory: /flyer/ ================== File [added]: etk-widescreen.png Delta lines: +0 -0 =================================================================== (Binary files differ) Property changes on: flyer/etk-widescreen.png ___________________________________________________________________

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 312)
by jfaber＠einsteintoolkit.org 13 Mar '12

13 Mar '12

User: jfaber Date: 2012/03/13 02:10 PM Modified: / ET.tex Log: Modified stress-energy source terms in initial data, vacuum evolution, anc constraint treatment to be in uniform notation. PLEASE DOUBLECHECK THE REVISED BSSN EQUATIONS BEFORE SUBMITTING! File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +17 -15 =================================================================== --- ET.tex 2012-03-13 18:18:06 UTC (rev 311) +++ ET.tex 2012-03-13 19:10:07 UTC (rev 312) @@ -1214,20 +1214,20 @@ \codename{Meudon\_Bin\_NS} handles binary NS data described in~\cite{Gourgoulhon:2000nn}, which represent solutions of the equations \begin{eqnarray*} -&&\nabla^2\nu_{(m)} = 4\pi\psi^4(\hat{E}_{(m)}+\hat{S}_{(m)})+\psi^4 K_{ij}K^{ij}_{(m)}-\nabla_i\nu_{(m)}\nabla^i\beta\\ -&&\nabla^2\beta_{(m)}=4\pi\psi^4\hat{S}_{(m)}+\frac{3}{4}\psi^4 K_{ij}K^{ij}_{(m)}-\frac{1}{2}(\nabla_i\nu_{(m)}\nabla^i\nu+\nabla_i\beta_{(m)}\nabla^i\beta)\\ -&&\nabla^2\beta^i_{(m)}+\frac{1}{3}\nabla^i\nabla_j \beta^j_{(m)} = -16\pi\alpha\psi^4(\hat{E}_{(m)}+P_{(m)})v^i_{(m)}+2\alpha\psi^4K^{ij}_{(m)}\nabla_j(3\beta-4\nu) +&&\nabla^2\nu_{(m)} = 4\pi\psi^4(E_{(m)}+S_{(m)})+\psi^4 K_{ij}K^{ij}_{(m)}-\nabla_i\nu_{(m)}\nabla^i\beta\\ +&&\nabla^2\beta_{(m)}=4\pi\psi^4S_{(m)}+\frac{3}{4}\psi^4 K_{ij}K^{ij}_{(m)}-\frac{1}{2}(\nabla_i\nu_{(m)}\nabla^i\nu+\nabla_i\beta_{(m)}\nabla^i\beta)\\ +&&\nabla^2\beta^i_{(m)}+\frac{1}{3}\nabla^i\nabla_j \beta^j_{(m)} = -16\pi\alpha\psi^4(E_{(m)}+P_{(m)})v^i_{(m)}+2\alpha\psi^4K^{ij}_{(m)}\nabla_j(3\beta-4\nu) \end{eqnarray*} where $\nu$ and $\beta$ are defined as \begin{equation} \nu\equiv \log\alpha;~~\beta\equiv\ln\alpha\psi^2. \end{equation} These equations are merely convenient reparameterizations of the ones used to generate binary BH data, with the matter source terms included. The extrinsic curvature is computed using~\ref{eq:lorextrcurv}, with both $K^{ij}$ and $\beta^i$ replaced by the split versions. -The matter sources terms are, respectively, -\begin{eqnarray} -\hat{E}&=&\alpha^2(u^0)^2 \rho h-P\\ -\hat{S}&=&3P+(\hat{E}+P)\frac{\alpha^2(u^0)^2 - 1}{\alpha^2(u^0)^2} -\end{eqnarray} +The matter sources terms $E$ and $S$, representing projections of the stress-energy tensor, are defined in~\ref{eq:e_tmunu} and \ref{eq:s_tmunu} below. +%\begin{eqnarray} +%e&=&n_\alpha n_\beta T^{\alpha\beta}=\alpha^2(u^0)^2 \rho h-P\\ +%S&=&3P+(\hat{E}+P)\frac{\alpha^2(u^0)^2 - 1}{\alpha^2(u^0)^2} +%\end{eqnarray} Lorene allows for two different NS spin states, either irrotational or synchronized. In the synchronized case, the velocity may be specified as a function of position once the orbital velocity is determined, while the irrotational case yields a rather complicated differential equation for the velocity potential which may then be used to determine the corresponding 3-velocity (see Equation~38 of \cite{Gourgoulhon:2000nn}). \codename{Meudon\_Mag\_NS} @@ -1277,7 +1277,7 @@ \\ \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 + \left( \tilde{A}^{ij} \tilde{A}_{ij} + \frac{1}{3} K^2 \right) + \frac{\alpha}{2}(E+ S) \\ \partial_0 \beta^i & = & \alpha^2 G(\alpha,\phi,x^\mu) B^i \\ @@ -1298,7 +1298,7 @@ & & {} + \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} + - \alpha e^{-4\phi} S_{ij}^{TF} \\ \partial_0\tilde{\Gamma}^i & = & \tilde{\gamma}^{kl}\partial_k\partial_l\beta^i @@ -1310,18 +1310,20 @@ + 2\alpha\left[ (m-1)\partial_k\tilde{A}^{ki} - \frac{2m}{3}\tilde{D}^i K \right. \nonumber \\ & & {} + m(\tilde{\Gamma}^i_{\; kl}\tilde{A}^{kl} + - 6\tilde{A}^{ij}\partial_j\phi) \Bigg] - S^i, + 6\tilde{A}^{ij}\partial_j\phi) \Bigg] -2\alpha \tilde{\gamma}^{ij} S_j \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} - E & \equiv & \frac{1}{\alpha^2} \left( T_{00} - 2 \beta^i T_{0i} + - \beta^i \beta^j T^{ij} \right) + E & \equiv & n_\alpha n_\beta T^{\alpha\beta} = \frac{1}{\alpha^2} \left( T_{00} - 2 \beta^i T_{0i} + + \beta^i \beta^j T^{ij} \right)\label{eq:e_tmunu} + \\ + S_{ij} & \equiv & \gamma_{i\alpha} \gamma_{j\beta} T^{\alpha\beta} \\ - S & \equiv & g^{ij} T_{ij} + S&\equiv&S^i_i = \gamma^{ij} S_{ij} \label{eq:s_tmunu} \\ - S_i & \equiv & - \frac{1}{\alpha} \left( T_{0i} - \beta^j T_{ij} \right) . + S_i & \equiv &-\gamma_{i\alpha} n_\beta T^{\alpha\beta}= - \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 involve

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 311)
by knarf＠cct.lsu.edu 13 Mar '12

13 Mar '12

User: knarf Date: 2012/03/13 01:18 PM Added: / iop_revised.tar.gz Removed: / arXiv-v2.tar.gz Log: add tarball uploaded to IOP as revised version Directory Changes: Directory: /svn:mime-type/ ========================== + application/octet-stream File Changes: Directory: / ============ File [removed]: arXiv-v2.tar.gz Delta lines: +0 -0 =================================================================== (Binary files differ) File [added]: iop_revised.tar.gz Delta lines: +0 -0 =================================================================== (Binary files differ) Property changes on: iop_revised.tar.gz ___________________________________________________________________

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[svn:einsteintoolkit] HydroBase/trunk/ (Rev. 48)
by bcmsma＠astro.rit.edu 13 Mar '12

13 Mar '12

User: bmundim Date: 2012/03/13 12:09 PM Modified: /trunk/ interface.ccl /trunk/doc/ documentation.tex Log: epsilon: specific internal energy (ie energy/mass) File Changes: Directory: /trunk/doc/ ====================== File [modified]: documentation.tex Delta lines: +1 -1 =================================================================== --- trunk/doc/documentation.tex 2012-02-04 15:49:06 UTC (rev 47) +++ trunk/doc/documentation.tex 2012-03-13 17:09:43 UTC (rev 48) @@ -137,7 +137,7 @@ \begin{itemize} \item \verb|rho|: rest mass density $\varrho$ \item \verb|press|: pressure $p$ - \item \verb|eps|: internal energy $\epsilon$ + \item \verb|eps|: specific internal energy $\epsilon$ \item \verb|vel[3]|: contravariant fluid three velocity $v^i$ defined as \begin{equation} v^i = \frac{u^i}{\alpha u^0} + \frac{\beta^i}{\alpha} Directory: /trunk/ ================== File [modified]: interface.ccl Delta lines: +1 -1 =================================================================== --- trunk/interface.ccl 2012-02-04 15:49:06 UTC (rev 47) +++ trunk/interface.ccl 2012-03-13 17:09:43 UTC (rev 48) @@ -31,7 +31,7 @@ CCTK_REAL rho type = GF Timelevels = 3 tags='ProlongationParameter="HydroBase::prolongation_type" tensortypealias="Scalar" interpolator="matter"' "rest mass density" CCTK_REAL press type = GF Timelevels = 3 tags='ProlongationParameter="HydroBase::prolongation_type" tensortypealias="Scalar" interpolator="matter"' "gas pressure" -CCTK_REAL eps type = GF Timelevels = 3 tags='ProlongationParameter="HydroBase::prolongation_type" tensortypealias="Scalar" interpolator="matter"' "internal energy density" +CCTK_REAL eps type = GF Timelevels = 3 tags='ProlongationParameter="HydroBase::prolongation_type" tensortypealias="Scalar" interpolator="matter"' "specific internal energy" CCTK_REAL vel[3] type = GF Timelevels = 3 tags='ProlongationParameter="HydroBase::prolongation_type" tensortypealias="U" interpolator="matter"' "velocity v^i"

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[svn:einsteintoolkit] GRHydro/trunk/ (Rev. 315)
by bcmsma＠astro.rit.edu 13 Mar '12

13 Mar '12

User: bmundim Date: 2012/03/13 12:06 PM Modified: /trunk/ param.ccl /trunk/doc/ documentation.tex Log: epsilon: specific internal energy (ie energy/mass) File Changes: Directory: /trunk/doc/ ====================== File [modified]: documentation.tex Delta lines: +6 -6 =================================================================== --- trunk/doc/documentation.tex 2012-02-16 23:18:45 UTC (rev 314) +++ trunk/doc/documentation.tex 2012-03-13 17:06:08 UTC (rev 315) @@ -204,11 +204,11 @@ For the equations of state, two ``types'' are recognized, controlled by the parameter {\tt GRHydro\_eos\_type}. These are {\tt "Polytype"} -where the pressure is a function of the density, $P=P(\rho)$, and the +where the pressure is a function of the rest-mass density, $P=P(\rho)$, and the more generic {\tt "General"} type where the pressure is a function -of the density and the internal energy, $P=P(\rho, \epsilon)$. For the +of the rest-mass density and the specific internal energy, $P=P(\rho, \epsilon)$. For the {\tt Polytype} equations of state one fewer equation is evolved and -the specific internal energy is set directly from the density. The +the specific internal energy is set directly from the rest-mass density. The actual equation of state used is controlled by the parameter {\tt GRHydro\_eos\_table}. Polytype equations of state include {\tt "2D\_Polytrope"} and general equations of state include {\tt @@ -312,8 +312,8 @@ $D$ is the generalized particle number density, $S^i$ are the generalized momenta in each direction, and $\tau$ is an internal energy term. These conserved variables are composed from a set of {\it primitive variables}, -which are $\rho$, the density, $p$, the -pressure, $v^i$, the fluid 3-velocities, $\epsilon$, the internal +which are $\rho$, the rest-mass density, $p$, the +pressure, $v^i$, the fluid 3-velocities, $\epsilon$, the specific internal energy, and $W$, the Lorentz factor, via the following relations % from GRHydro/src/Prim2con.F90 % w = 1.d0 / sqrt(1.d0 - (gxx*dvelx*dvelx + gyy*dvely*dvely + gzz & @@ -342,7 +342,7 @@ Only five of the primitive variables are independent. Usually the Lorentz factor is defined in terms of the velocities and the metric as $W = (1-\gamma_{ij}v^i v^j)^{-1/2}$. -Also one of the pressure, density or internal energy terms is given in +Also one of the pressure, rest-mass density or specific internal energy terms is given in terms of the other two by an {\it equation of state}. The fluxes are usually defined in terms of both the conserved Directory: /trunk/ ================== File [modified]: param.ccl Delta lines: +1 -1 =================================================================== --- trunk/param.ccl 2012-02-16 23:18:45 UTC (rev 314) +++ trunk/param.ccl 2012-03-13 17:06:08 UTC (rev 315) @@ -322,7 +322,7 @@ 0: :: "greater than zero" } 1.e-20 -real GRHydro_eps_min "Minimum value of internal energy - this is now only used in GRHydro_InitData's GRHydro_Only_Atmo routine" +real GRHydro_eps_min "Minimum value of specific internal energy - this is now only used in GRHydro_InitData's GRHydro_Only_Atmo routine" { 0: :: "Positive" } 1.e-10

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 310)
by jfaber＠einsteintoolkit.org 13 Mar '12

13 Mar '12

User: jfaber Date: 2012/03/13 11:19 AM Modified: / ET.tex Log: One more change to "specific internal energy" File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +2 -2 =================================================================== --- ET.tex 2012-03-13 16:15:16 UTC (rev 309) +++ ET.tex 2012-03-13 16:19:25 UTC (rev 310) @@ -1519,8 +1519,8 @@ hydrodynamic quantities, with several different reconstruction methods and Riemann solvers, as we discuss below. In such a scheme, we define a set of ``conserved'' hydrodynamic variables, defined in terms of the -``primitive'' physical variables such as mass and internal energy -density, pressure, and velocity. Wherever derivatives of hydrodynamic +``primitive'' physical variables such as mass density, specific internal energy, +pressure, and velocity. Wherever derivatives of hydrodynamic terms appear in the evolution equations for the conserved variables, they are restricted to appear only inside divergence terms (referred to as fluxes) and never in the source terms. By calculating fluxes between pairs of neighboring points where field and hydrodynamic terms are evaluated, we

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 309)
by jfaber＠einsteintoolkit.org 13 Mar '12

13 Mar '12

User: jfaber Date: 2012/03/13 11:15 AM Modified: / ET.tex Log: Fixed the description in to TOVSolver footnote about notation; no footnote necessary File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +5 -5 =================================================================== --- ET.tex 2012-03-13 15:14:05 UTC (rev 308) +++ ET.tex 2012-03-13 16:15:16 UTC (rev 309) @@ -710,7 +710,7 @@ $K_{ij}$ & Extrinsic curvature & \protect\ref{eq:extrcurv} & ADMBase::curv\\ $\rho$ & Rest mass density & \protect\ref{eq:Tmunu},\protect\ref{eq:enthalpy} & HydroBase::rho\\ $P$ & Fluid pressure & \protect\ref{eq:enthalpy} & HydroBase::press\\ -$\epsilon$ & Internal energy density & \protect\ref{eq:enthalpy} & Hydrobase::eps\\ +$\epsilon$ & Specific internal energy & \protect\ref{eq:enthalpy} & Hydrobase::eps\\ $h$ & Specific enthalpy & \protect\ref{eq:enthalpy} & N/A\\ $v^i$ & 3-velocity & \protect\ref{eq:3vel} & HydroBase::vel\\ $B^i$ & Magnetic field vector & \protect\ref{eq:Bi} & HydroBase::Bvec\\ @@ -884,7 +884,7 @@ \begin{itemize} \item \verb|rho|: rest mass density $\rho$ \item \verb|press|: pressure $P$ - \item \verb|eps|: internal energy density $\epsilon$ + \item \verb|eps|: Specific internal energy $\epsilon$ \item \verb|vel[3]|: contravariant fluid three velocity $v^i$ defined as \begin{equation} v^i = \frac{u^i}{\alpha u^0} + \frac{\beta^i}{\alpha}\label{eq:3vel} @@ -1116,14 +1116,14 @@ radius $\hat{r}$: \begin{eqnarray} \label{eq:TOViso} - \frac{d P}{d \hat{r}} & = & -(e + P) \frac{M_e + 4\pi \hat{r}^3 P}{\hat{r}(\hat{r} - 2M_e)}\nonumber\\ + \frac{d P}{d \hat{r}} & = & -(\mu + P) \frac{M_e + 4\pi \hat{r}^3 P}{\hat{r}(\hat{r} - 2M_e)}\nonumber\\ % - \frac{d M_e}{d \hat{r}} & = & 4 \pi \hat{r}^2 e\nonumber\\ + \frac{d M_e}{d \hat{r}} & = & 4 \pi \hat{r}^2 \mu\nonumber\\ % \frac{d \Phi}{d \hat{r}} & = & \frac{M_e + 4\pi \hat{r}^3 P}{\hat{r}(\hat{r} - 2M_e)}. \end{eqnarray} -where $e\equiv \rho(1+\epsilon)$ is the energy density of the fluid, including the internal energy contribution\footnote[1]{We note that since different application thorns may define their own local variables, the energy density is referred to as {\tt rho} within \codename{TOVSolver}, as the projected energy density $E$, defined in Sec.~\protect\ref{sec:Kevol}, is within \codename{McLachlan} and a few other thorns. Similar ambiguities exist for other commonly used variable names, particularly $\phi$ and $\alpha$.}. +where $\mu\equiv \rho(1+\epsilon)$ is the energy density of the fluid, including the internal energy contribution. The routine also supplies the analytically known solution in the exterior, \begin{eqnarray}

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 308)
by knarf＠cct.lsu.edu 13 Mar '12

13 Mar '12

User: knarf Date: 2012/03/13 10:14 AM Added: / arXiv-v1.tar.gz, arXiv-v2.tar.gz Removed: / arXiv.tar.gz Log: rename old arXiv version to v1 and add proposed new version as v2 Directory Changes: Directory: /svn:mime-type/ ========================== + application/octet-stream File Changes: Directory: / ============ File [added]: arXiv-v1.tar.gz Delta lines: +0 -0 =================================================================== (Binary files differ) File [added]: arXiv-v2.tar.gz Delta lines: +0 -0 =================================================================== (Binary files differ) Property changes on: arXiv-v2.tar.gz ___________________________________________________________________ File [removed]: arXiv.tar.gz Delta lines: +0 -0 =================================================================== (Binary files differ)

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[svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 307)
by knarf＠cct.lsu.edu 13 Mar '12

13 Mar '12

User: knarf Date: 2012/03/13 09:49 AM Modified: / ET.tex Log: fix newly introduced typos File Changes: Directory: / ============ File [modified]: ET.tex Delta lines: +6 -6 =================================================================== --- ET.tex 2012-03-13 14:41:53 UTC (rev 306) +++ ET.tex 2012-03-13 14:49:04 UTC (rev 307) @@ -1195,7 +1195,7 @@ \end{figure} \codename{Meudon\_Bin\_BH} can read in BH-BH binary initial data described -in~\cite{Grandclement:2001ed}, representing solutions to the hamiltonian and momentum constraints, along with the trace of the spatial components of the Einstein equations, form the linked elliptic equation set: +in~\cite{Grandclement:2001ed}, representing solutions to the Hamiltonian and momentum constraints, along with the trace of the spatial components of the Einstein equations, form the linked elliptic equation set: \begin{eqnarray*} &&\nabla^2\alpha_{(m)} = \alpha \psi^4 K_{ij} K^{ij}_{(m)}\\ &&\nabla^2\psi_{(m)}=-\frac{\psi^5}{8}K_{ij}K^{ij}_{(m)}\\ @@ -2081,8 +2081,8 @@ \label{eq:christodoulou} \end{equation} that takes into account the -contributions of the irreducuble mass $M_{\mathrm{ir}}$ and angular -momemtum $S$ to the total black hole mass $M$. +contributions of the irreducible mass $M_{\mathrm{ir}}$ and angular +momentum $S$ to the total black hole mass $M$. Finally, the module \codename{HydroAnalysis} additionally locates the (coordinate) center of mass as well as the point of maximum rest mass density of a @@ -2496,7 +2496,7 @@ extracted at $R=30\mathrm{M}$, and its numerical convergence. In this and in the following sections, a numerical quantity $\Psi$ is said to converge with convergence order $Q$ if for a set of numerical -stepsizes $h_1$, $h_2$, $h_3$ the difference between successive +step sizes $h_1$, $h_2$, $h_3$ the difference between successive resolutions scales as \begin{equation} \frac{\Psi_{h_1}-\Psi_{h_2}}{\Psi_{h_2}-\Psi_{h_3}} = \frac{h_1^Q-h_2^Q}{h_2^Q-h_3^Q}\,. @@ -3021,8 +3021,8 @@ star's coordinate radius is twice the size of the forming apparent horizon~\cite{Thierfelder:2010dv}. At this point the majority of the matter is inside the apparent horizon, whose size quickly grows to its -final size while the numerical spacetime aproaches the final ``trumpet'' -configuraton. +final size while the numerical spacetime approaches the final ``trumpet'' +configuration. In Figure~\ref{fig:tov_collapse_H_convergence_at0}, we display the convergence factor for the Hamiltonian constraint

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