[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 252)
knarf at cct.lsu.edu
knarf at cct.lsu.edu
Wed Feb 29 21:40:21 CST 2012
User: knarf
Date: 2012/02/29 09:40 PM
Modified:
/
ET.tex
Log:
addressed more reviewer comments
File Changes:
Directory: /
============
File [modified]: ET.tex
Delta lines: +28 -21
===================================================================
--- ET.tex 2012-02-17 17:41:49 UTC (rev 251)
+++ ET.tex 2012-03-01 03:40:20 UTC (rev 252)
@@ -816,6 +816,7 @@
\subsubsection{TmunuBase}
In the Einstein Toolkit, we typically choose the stress energy tensor $T^{\mu\nu}$ to be that of an ideal relativistic fluid,
\begin{equation}
+\label{eq:Tmunu}
T^{\mu\nu} = \rho h u^\mu u^\nu + g^{\mu\nu} P\,\,,
\end{equation}
where $\rho$, $u^\mu$, and
@@ -1035,8 +1036,13 @@
\frac{d m}{d r} & = & 4 \pi r^2 e\nonumber\\
%
\frac{d \Phi}{d r} & = & \frac{m + 4\pi r^3 P}{r(r -
- 2m)},
+ 2m)}.
\end{eqnarray}
+This assumes the stress energy tensor is given by
+\begin{equation}
+ \label{eq:Tmunu_e}
+ T^{\mu\nu} = (e + P)u^{\mu}u^{\nu} + Pg^{\mu\nu},
+\end{equation}
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 $\alpha$.}, $m$ is the gravitational mass inside a sphere of radius $r$, and
$\Phi$ the logarithm of the lapse. The routine also supplies the analytically known
solution in the exterior,
@@ -1046,7 +1052,7 @@
\Phi & = &\dfrac{1}{2} \log(1-2M / r)
\label{eq:TOVexterior}
\end{eqnarray}
-where {\tt TOV\_atmosphere} is a parameter used to define the density of the
+where {\tt TOV\_atmosphere} is a parameter used to specify the density of the
ambient atmosphere. 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
@@ -1096,8 +1102,8 @@
\\
K & \equiv & g^{ij} K_{ij}\,,
\\
- \tilde A_{ij} & \equiv & e^{-4\phi} \left[ K_{ij} - \frac{1}{3} g_{ij} K\,,
- \right]
+ \tilde A_{ij} & \equiv & e^{-4\phi} \left[ K_{ij} - \frac{1}{3} g_{ij} K
+ \right]\,,
\\
\tilde\Gamma^i & \equiv & \tilde\gamma^{jk} \tilde\Gamma^i_{jk} .
\end{eqnarray}
@@ -1182,8 +1188,8 @@
\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 $1+\log$
+related to the time derivative of the shift. The gauge functions $f$,
+$K_0$, $G$, $H$, and $\eta$ are determined by our choice of a $1+\log$
\cite{Alcubierre:2002kk} slicing:
\begin{eqnarray}
f(\alpha,\phi,x^\mu) & \equiv & 2/\alpha
@@ -1194,7 +1200,7 @@
\begin{eqnarray}
G(\alpha,\phi,x^\mu) & \equiv & (3/4)\, \alpha^{-2}
\\
- H(\alpha,\phi,x^\mu) & \equiv & \exp\{4\phi\}
+ H(\alpha,\phi,x^\mu) & \equiv & \e^{4\phi}
\\\label{eq:eta}
\eta(B^i,\alpha,x^\mu) & \equiv & (1/2)\, B^i q(r) .
\end{eqnarray}
@@ -1216,7 +1222,8 @@
evolve only very slowly, e.g., far away from the source.
Here we use the simple damping mechanism described in (12) of
-\cite{Schnetter:2010cz}, which is defined as:
+\cite{Schnetter:2010cz}, through defining the function $q$
+in~\eref{eq:eta}:
\begin{eqnarray}
\label{eq:varying-simple}
q(r) & \equiv & \left\{
@@ -1229,11 +1236,9 @@
\end{eqnarray}
with a constant $R$ defining the transition radius between the
interior, where $q\approx1$, and the exterior, where $q$ falls off as
-$1/r$. A description of how $q$ appears in the gauge
-parameters may be found in~\eref{eq:eta}.
+$1/r$.
-
\subsubsection{Initial Conditions}
Initial conditions for the ADM variables $g_{ij}$, $K_{ij}$, lapse
@@ -1242,7 +1247,7 @@
quantities are calculated via their definitions, 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
+are calculated from spatial derivatives of the metric, and one cannot use
centered finite differencing stencils near the outer boundary.
The extrapolation stencils distinguish between points on the faces,
@@ -1381,7 +1386,9 @@
\end{equation}
where $ \nabla_{\!\mu} $ denotes the covariant derivative with respect
to the 4-metric, and $ J^{\,\mu} = \rho u^{\,\mu} $ is the mass current.
-%with the 4-velocity $ u^{\,\mu} $ and the rest-mass density $\rho$. $
+$T^{\mu \nu}$ 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
@@ -1466,7 +1473,7 @@
different forms of the slope limiter available. In practice, all
try to accomplish the same task of preserving monotonicity and removing
the possibility of spuriously creating local extrema. Implemented methods
-include minmod, superbee~\cite{Roe:1986cb}, and monotonized central~\cite{vanLeer:1977aa}.
+include minmod, superbee (both~\cite{Roe:1986cb}), and monotonized central~\cite{vanLeer:1977aa}.
The piecewise parabolic method (PPM) is a multi-step method based around
a quadratic fit to nearby points interpolated to cell faces
@@ -1549,7 +1556,7 @@
\begin{equation}
f(q)=\frac{1}{2}\left(f(q^L)+f(q^R)-\sum |\lambda_i| \Delta w_i r^i\right)
\end{equation}
-where the eigenvector appearing in the summed term are evaluated for
+where the eigenvectors appearing in the summed term are evaluated for
the approximate Roe average flux $q_{\rm Roe}=\frac{1}{2}(q^L+q^R)$.
The Marquina flux routines use a similar approach to the Roe method,
but provide a more accurate treatment for supersonic flows (i.e., those
@@ -1634,7 +1641,7 @@
d\ln{\rho}$ is related to the frequently used polytropic index $n$ via
$n = 1 / (\Gamma - 1)$.
-The gamma-law EOS\footnote{For historic reasons, this EOS is referred to
+The gamma-law EOS\footnote{For historical reasons, this EOS is referred to
as the ``ideal fluid'' EOS in \codename{GRHydro}.},
\begin{equation}
P = (\Gamma - 1) \rho \epsilon\,\,,
@@ -1644,11 +1651,11 @@
used extensively in simulations of NS-NS and BH-NS mergers.
The hybrid EOS, first introduced by~\cite{Janka:1993da}, is a
-2-piecewise polytropic with a thermal component designed for the
+2-piecewise polytrope with a thermal component designed for the
application in simple models of stellar collapse. At densities below
nuclear density, a polytropic EOS with $\Gamma = \Gamma_1 \approx 4/3$ is
used. To mimic the stiffening of the nuclear EOS at nuclear density,
-the low-density polytrope is fitted to a second polytrope with
+the low-density polytrope is matched to a second polytrope with
$\Gamma = \Gamma_2 \gtrsim 2$. To allow for thermal contributions to the
pressure due to shock heating, a gamma-law with $\Gamma = \Gamma_\mathrm{th}$
is used. The full EOS then reads
@@ -1675,7 +1682,7 @@
\codename{EOS\_Omni} also integrates the \codename{nuc\_eos} driver
routine, which was first developed for the \codename{GR1D} code
\cite{O'Connor:2009vw} for tabulated microphysical finite-temperature EOS
-which assume nuclear statistical equilibrium (NSE). \codename{nuc\_eos}
+which assume nuclear statistical equilibrium. \codename{nuc\_eos}
handles EOS tables in \codename{HDF5} format which contain entries for
thermodynamic variables $X = X(\rho,T,Y_e)$, where $T$ is the matter
temperature and $Y_e$ is the electron fraction. \codename{nuc\_eos}
@@ -1726,12 +1733,12 @@
In \codename{EHFinder}, the null surface is represented by a function
$f(t,x^i)=0$ which is required to satisfy the null condition
-$g^{\alpha\beta} \partial_\alpha f \partial_\beta f = 0$. In the
+$g^{\alpha\beta} (\partial_\alpha f) (\partial_\beta f) = 0$. In the
standard numerical 3+1 form of the metric, this null condition can be
expanded out into an evolution equation for $f$ as
\begin{equation}
\partial_t f = \beta^i \partial_i f - \sqrt{\alpha^2 \gamma^{ij}
- \partial_i f \partial_j f}
+ (\partial_i f) (\partial_j f)}
\end{equation}
where the roots are chosen to describe outgoing null geodesics. The
function $f$ is chosen such that it is negative within the initial
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