Added Miguels docs for radiation boundary conditions from the source files

git-svn-id: http://svn.cactuscode.org/arrangements/CactusBase/Boundary/trunk@124 6a38eb6e-646e-4a02-a296-d141613ad6c4

1 files changed, 52 insertions, 3 deletions

diff --git a/doc/documention.tex b/doc/documention.tex
index 356b298..aa1874d 100644
--- a/doc/documention.tex
+++ b/doc/documention.tex
@@ -169,15 +169,64 @@ boundary to apply the boundary condition.
 
 \subsection{Radiation Boundary Condition}
 
-This is a two level scheme. Specify grid functions on the current time 
-level (to which the BC is applied) and grid functions from a past
-timelevel.
+This is a two level scheme. Grid functions are given for the current time 
+level (to which the BC is applied) as well as grid functions from a past
+timelevel which are needed for constructing the boundaray condition.
 The grid function of the past time level need to have the same
 geometry. When applying this boundary condition to a group, the
 members of the group have to match up. Currently radiative boundary
 conditions can only be applied with a stencil width of one in each
 direction. 
 
+The radiative boundary condition that is implemented is
+\begin{equation}
+\label{eqrad}
+f = f_0 + \frac{u(r-vt)}{r}+\frac{h(r+vt)}{r}
+\end{equation}
+That is, outgoing radial waves with a 1/r
+fall off, and the correct asymptotic value f0 are assumed, including
+the possibility of incoming waves
+(these incoming waves should be modeled somehow).
+
+Condition~\ref{eqrad} above leads to the differential equation:
+\begin{equation}
+\frac{x^i}{r}\frac{\partial f}{\partial t}
++ v \frac{\partial f}{\partial x^i}
++\frac{v x^i}{r^2} (f-f_0)
+= H \frac{v x^i}{r^2}  
+\end{equation}
+where $x^i$ is the normal direction to the given boundaries,
+and $H = 2 dh(s)/ds$.
+
+At a given boundary only the derivatives in the normal direction are 
+considered.  Notice that $u(r-vt)$ has disappeared, but we still do 
+not know the value of $H$.
+
+To get $H$ we do is the following:  The expression is evaluated one 
+point in from the boundary and solved for $H$ there. Now need a way of 
+extrapolating $H$ to the boundary is required. For this, assume that 
+$H$ falls off as a power law:
+\begin{equation}
+H = \frac{k}{r^n} \qquad \mbox{which gives} \qquad d_i H  =  - n \frac{H}{r}
+\end{equation}
+The value of $n$ is is defined by the parameter {\tt radpower}.
+If this parameter is negative, $H$ is forced to be zero (this
+corresponds to pure outgoing waves and is the default).
+
+The observed behaviour is the following:  Using $H=0$
+is very stable, but has a very bad initial transient. Taking
+$n$ to be 0 or positive improves the initial behaviour considerably,
+but introduces a drift that can kill an evolution at very late
+times.  Empirically, the best value found so far is $n=2$, for
+which the initial behaviour is very nice, and the late time drift 
+is quite small.
+
+Another problem with this condition is that it does not
+use the physical characteristic speed, but rather it assumes
+a wave speed of $v$, so the boundaries should be out in
+the region where the characteristic speed is constant.
+Notice that this speed does not have to be 1.  
+
 \subsubsection*{Calling from C:}
 \begin{verbatim}
 int ierr = BndRadiativeVN(cGH *cctkGH, int *stencil_size,