[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 289)

[jfaber at einsteintoolkit.org]

User: jfaber
Date: 2012/03/12 11:40 AM

Modified:
 /
  ET.tex

Log:
 Notation table margin fixed
 other todo's handled, except for overall style review

File Changes:

Directory: /
============

File [modified]: ET.tex
Delta lines: +15 -15
===================================================================
--- ET.tex	2012-03-12 16:35:53 UTC (rev 288)
+++ ET.tex	2012-03-12 16:40:21 UTC (rev 289)
@@ -490,11 +490,11 @@
 step is called \emph{regridding}.
 
 Is most simulations using Carpet the sizes of refinement levels are
-predescribed by the user, while their locations are adapted according
-to the locations of interesting features such as black holes or stars.
-In addition, refinement levels are often activated is disabled depending
-on simulation dynamics, such as the formation of a common horizon, or
-an increasing stellar density due to collaps.
+predetermined by the user, while their locations are calculated adaptively
+to track the locations of interesting features such as black holes or stars.
+In addition, refinement levels may be disabled or activated depending
+on the dynamics of a particular simulation, responding, e.g.,  to events such  as the formation of a common horizon or
+an increasing stellar density due to collapse, respectively.
 
 Since a finer grid spacing also requires smaller time steps for
 hyperbolic problems, the finer grids perform multiple time steps for
@@ -696,12 +696,12 @@
 \label{table:vars}
 {\centering
 \begin{tabular}{llll}
-Symbol & Quantity & Equation & ET thorn::group \\
+Symbol & Quantity & Eqn. & ET thorn::group \\
 \hline
 $G_{\mu\nu}$ & Einstein Tensor & \protect\ref{eq:einstein} & N/A\\
 $T_{\mu\nu}$ & Stress-Energy Tensor & \protect\ref{eq:Tmunu} & TmunuBase::stress\_energy\_tensor\\
 $g_{\mu\nu}$ & Spacetime 4-metric & \protect\ref{eq:adm} & N/A\\
-$F_{\mu\nu},~(^*F^{\mu\nu})$ & (Dual) Faraday tensor & \protect\ref{eq:Bi} & N/A\\
+$F_{\mu\nu}$ & Faraday tensor & \protect\ref{eq:Bi} & N/A\\
 $u^\mu$ & 4-velocity & N/A & N/A\\
 $\gamma_{ij}$ & Spatial 3-metric & \protect\ref{eq:adm} & ADMBase::metric\\
 $\alpha$  & Lapse function & \protect\ref{eq:adm} & ADMBase::lapse\\
@@ -717,7 +717,7 @@
 $\psi$, ($\phi$) & (Logarithmic) conformal factor & \protect\ref{eq:confpsi},\protect\ref{eq:phi} & (ML\_BSSN/ML\_log\_confac) \\
 $\tilde{\gamma}_{ij}$ & Conformal 3-metric & \protect\ref{eq:tildegamma} & ML\_BSSN/ML\_metric \\
 $K$ & Trace of extrinsic curvature & \protect\ref{eq:trK} & ML\_BSSN/ML\_trace\_curv \\
-$\tilde{A}_{ij}$ & Conformal traceless extrinsic curvature & \protect\ref{eq:Atij} & ML\_BSSN/ML\_curv \\
+$\tilde{A}_{ij}$ & Conformal extrinsic curvature & \protect\ref{eq:Atij} & ML\_BSSN/ML\_curv \\
 $\tilde{\Gamma}^i$ & Conformal connection & \protect\ref{eq:Gammai} & ML\_BSSN/ML\_Gamma \\
 $D$ & Conservative density & \protect\ref{eq:p2c1} & GRHydro::dens\\
 $S^i$ & Conservative momentum & \protect\ref{eq:p2c2} & GRHydro::scon\\
@@ -1521,23 +1521,23 @@
 density, 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 at cell faces, we may
-obtain a consistent description of the inter-cell values using
-reconstruction techniques that account for the fact that hydrodynamic
-variables are not smooth and may not be finite differenced accurately.
+(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
+obtain a consistent description of time evolution using HRSC
+reconstruction techniques that accounts for the fact that hydrodynamic
+variables are not smooth and may not be finite differenced accurately \footnote{While the \codename{Carpet} AMR driver uses a so-called ``vertex-centered'' approach, \codename{GRHydro} is sufficiently general in its techniques that it may be used with either ``cell-centered'' or ``vertex-centered'' grid infrastructures; see, e.g., \protect\cite{Baumgarte:2010nu} for a review of these methods}.  
 All other source terms in the evolution equations may contain only the
 hydrodynamic variables themselves and the metric variables and derivatives of the latter, since the metric must formally be smooth and thus
 differentiable using finite differencing techniques.  Summarizing
 these methods briefly, the following stages occur every timestep:
 \begin{itemize}
-\item The primitive variables are ``reconstructed'' at cell faces
+\item The primitive variables are ``reconstructed'' between neighboring points
   using shock-capturing techniques, with total variation diminishing
   (TVD), piecewise parabolic (PPM), and essentially non-oscillatory
   (ENO) methods currently implemented.
-\item A Riemann problem is solved at each cell face using an
+\item A Riemann problem is solved to generate the fluxes corresponding to the reconstructed values using an
   approximate solver.  Currently implemented versions include HLLE
   (Harten-Lax-van Leer-Einfeldt), Roe, and Marquina solvers.
-\item The conserved variables are advanced one timestep, and used to
+\item Using the computed flux  and source terms, the conserved variables are advanced one timestep, and used to
   recalculate the new values of the primitive variables.
 \end{itemize}
 We discuss the GRHD formalism,  the stages within a timestep, and the