[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 68)

[cott at tapir.caltech.edu]

User: cott
Date: 2011/04/27 01:39 PM

Modified:
 /
  ET.tex

Log:
 * more great stuff in curvature evolution section

File Changes:

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

File [modified]: ET.tex
Delta lines: +11 -11
===================================================================
--- ET.tex	2011-04-27 18:24:03 UTC (rev 67)
+++ ET.tex	2011-04-27 18:39:57 UTC (rev 68)
@@ -696,6 +696,8 @@
 a central access point for analysis thorns.
 
 \subsection{Initial Data\pages{4 Josh/Bruno}}
+\label{sec:initial_data}
+
 The Einstein Toolkit contains many modules used to generate initial data for 
 general relativistic simulations, including both vacuum and hydrodynamical
 configurations.  
@@ -1131,16 +1133,14 @@
 
 \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.
+Initial conditions 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_data}. From these the BSSN
+quantities are calculated 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. 
 
 The extrapolation stencils distinguish between points on the faces,
 edges, and corners of the grid. Points on the faces are extrapolated
@@ -1150,7 +1150,7 @@
 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
+schemes (e.g., three layers for a fourth order scheme), one proceeds 
 outwards starting from the innermost layer. Each subsequent layer is
 then defined via the points in the interior and the previously
 calculated layers.