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