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.