From c5ceb72630d8f43a635315ac23089c0854799b1e Mon Sep 17 00:00:00 2001 From: allen Date: Thu, 7 Dec 2000 15:50:38 +0000 Subject: 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 --- doc/documention.tex | 55 ++++++++++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 52 insertions(+), 3 deletions(-) (limited to 'doc') 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, -- cgit v1.2.3