Describe "Minkowski/shifted gauge wave" and "Schwarzschild/BL".

git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/Exact/trunk@223 e296648e-0e4f-0410-bd07-d597d9acff87

1 files changed, 84 insertions, 3 deletions

diff --git a/doc/documentation.tex b/doc/documentation.tex
index ad6d338..3bc1950 100644
--- a/doc/documentation.tex
+++ b/doc/documentation.tex
@@ -80,7 +80,8 @@
 
 \author{Original code by Carsten Gundlach and Miguel Alcubierre, \\
         exact solutions added by many other people,	\\
-	this documentation by Jonathan Thornburg}
+	this documentation by Jonathan Thornburg,
+        and by other people}
 
 % The title of the document (not necessarily the name of the Thorn)
 \title{Thorn Guide for the {\bf Exact} Thorn}
@@ -212,6 +213,8 @@ Model Name
 	& --	& Minkowski spacetime in non-trivial spatial coordinates\\
 {\tt "Minkowski/gauge wave"}
 	& --	& Minkowski spacetime in gauge-wave coordinates		\\
+{\tt "Minkowski/shifted gauge wave"}
+	& --	& Minkowski spacetime in shifted gauge-wave coordinates	\\
 {\tt "Minkowski/conf wave"}
 	& --	& Minkowski spacetime with $\sin$ in conformal factor	\\[1ex]
 %
@@ -222,6 +225,8 @@ Model Name
 {\tt "Schwarzschild/PG"}
 	& --	& Schwarzschild spacetime in Painlev\'{e}-Gullstrand
 		  coordinates (these have a flat 3-metric)		\\
+{\tt "Schwarzschild/BL"}
+	& --	& Schwarzschild spacetime in Brill-Lindquist coordinates\\
 {\tt "Schwarzschild/Novikov"}
 	& --	& Schwarzschild spacetime in Novikov coordinates	\\
 {\tt "Kerr/Boyer-Lindquist"}
@@ -599,8 +604,9 @@ The line element is
 \begin{equation}
 ds^2=-H dt^2 +Hdx^2+dy^2+dz^2,
 \end{equation}
-where $H=H(x-t)$, for instance $H=1-a\sin\left((x-t)/\Lambda\right)$.
-This is flat space but the slice is a planar wave travelling along the x axis.
+where $H=H(x-t)$, for instance $H=1-A\sin\left((x-t)/\Lambda\right)$.
+This is a flat spacetime, but the slice is a planar wave travelling
+along the x axis.
 
 This thorn implements several possible choices for the $H$ function,
 controlled by the \verb|Minkowski_gauge_wave__what_fn| parameter:
@@ -634,6 +640,51 @@ periodic boundary wouldn't make sence), especially in the diagonal case.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+\subsection{Minkowski Spacetime in shifted gauge-wave coordinates}
+
+\verb|Exact::exact_model = "Minkowski/shifted gauge wave"| specifies Minkowski
+spacetime with the ``shifted gauge-wave'' coordinates suggested by
+Jeff Winicour:
+The line element is
+\begin{equation}
+ds^2 = (H-1)\, dt^2 + (H+1)\, dx^2 - 2H\, dt\, dx + dy^2 + dz^2
+\end{equation}
+where $H=H(x-t)$, for instance $H=A\sin\left((x-t)/\Lambda\right)$.
+This is a flat spacetime, but the slice is a planar wave travelling
+along the x axis.
+
+This thorn implements one choice for the $H$ function, controlled by
+the \verb|Minkowski_gauge_wave__what_fn| parameter:
+\begin{eqnarray}
+H(x-t)	&=& 1 - A \sin \left(\frac{x-\omega t}{\Lambda} - \delta\right)
+\end{eqnarray}
+
+The parameters are
+\begin{itemize}
+\item	$A = \verb|Minkowski_gauge_wave__amplitude|$, the amplitude
+\item	$\omega = \verb|Minkowski_gauge_wave__omega|$, the angular frequency
+\item	$\lambda = \verb|Minkowski_gauge_wave__lambda|$, the wavelength
+\item	$\delta = \verb|Minkowski_gauge_wave__phase|$, the phase shift
+\end{itemize}
+A plane wave has $\omega = \pm \lambda$ for a wave that travels in the
+$x$ direction, and $\omega = \pm \lambda \sqrt{2}$ for a diagonal
+wave.
+
+If the Boolean parameter \verb|Minkowski_gauge_wave__diagonal| is
+true, then we make the gauge wave travel diagonally across the grid by
+the coordinate transformation
+\begin{eqnarray}
+x &=& \frac{1}{\sqrt{2}}(x^\prime - y^\prime)				\\
+y &=& \frac{1}{\sqrt{2}}(x^\prime + y^\prime)				%%%\\
+\end{eqnarray}
+For code testing, the idea is to test evolving this with periodic
+boundary conditions, to see whether the code is able to cope with
+that.  The tricky part is to make the wave fit the grid exactly
+(otherwise the periodic boundary wouldn't make sence), especially in
+the diagonal case.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
 \subsection{Minkowski Spacetime with $\sin$ term in conformal factor}
 
 \verb|Exact::exact_model = "Minkowski/conf wave"| specifies Minkowski
@@ -789,6 +840,36 @@ g_{ab} = \left[
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+\subsection{Schwarzschild spacetime in Brill--Lindquist coordinates}
+
+\verb|Exact::exact_model = "Schwarzschild/BL"| specifies Schwarzschild
+spacetime in Brill--Lindquist coordinates.  These coordinates have the
+interesting property that the spatial metric is conformally flat and
+time-symmetric for the initial data.  The only physics parameter is
+the black hole mass $m = \verb|Schwarzschild_BL__mass|$.
+
+There is also a numerical parameter \verb|Schwarzschild_BL__epsilon|
+which is used to avoid division by zero if a grid point falls exactly
+at the origin; the default setting should be ok for most purposes.
+
+In the usual Cactus $(t,x,y,z)$ Cartesian-topology coordinates, the
+4-metric is given by
+\begin{eqnarray}
+   \alpha & = & 1
+\\
+   \beta^i & = & 0
+\\
+   \gamma_{ij} & = & \Psi^4\, \delta_{ij}
+\end{eqnarray}
+with the conformal factor
+\begin{eqnarray}
+   \Psi & = & 1 + \frac{m}{2r}
+\end{eqnarray}
+where $r$ is the coordinate radius.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
 \subsection{Schwarzschild Spacetime in Novikov coordinates}
 
 \verb|Exact::exact_model = "Novikov"| specifies the unit-mass Schwarzschild