\documentclass{article}

% Use the Cactus ThornGuide style file
% (Automatically used from Cactus distribution, if you have a 
%  thorn without the Cactus Flesh download this from the Cactus
%  homepage at www.cactuscode.org)
\usepackage{../../../../doc/ThornGuide/cactus}

\begin{document}

\title{IDAxiBrillBH}
\author{Paul Walker, Steve Brandt,\\some cleanups by Jonathan Thornburg}
\date{$ $Date$ $}

\maketitle

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\begin{abstract}
  Thorn IDAxiBrillBH provides analytic initial data for a vacuum
  black hole spacetime:  a single Schwarzschild black hole in
  isotropic coordinates plus Brill wave.  This initial data is
  provided for the 3-conformal metric, it's spatial derivatives, and
  extrinsic curvature.
\end{abstract}

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\section{Purpose}

The pioneer, Bernstein, studied a single black hole which is
non-rotating and distorted in azimuthal line symmetry of 2 dimensional 
case \cite{Bernstein93a}. In this non-rotating case, one chooses the
condition, $K_{ij} = 0$,  and 
\begin{equation}
\gamma_{ab} = \psi^4 \hat \gamma_{ab},
\end{equation}
where $\gamma_{ab}$ is the physical three metric and
$\hat{\gamma}_{ab}$ is some chosen conformal three metric.

The Hamiltonian constraint reduces to
\begin{equation}
\hat \Delta \psi = \frac{1}{8}\psi \hat R,
\label{IDAxiBrillBH/eqn:conformal-hamiltonian}
\end{equation}
where $\hat \Delta$ is the covariant Laplacian and $\hat R$ is the
Ricci tensor for the conformal three metric. This form allows
us to choose an arbitrary conformal three metric, and then solve an
elliptic equation for the conformal factor, therefore satisfying the
constraint equations ($K_{ij} = 0$ trivially satisfies the momentum
constraints in vacuum).  This approach was used to create
``Brill waves'' in a spacetime without black holes \cite{Brill59}.
Bernstein extended this to the black hole spacetime.  Using
spherical-polar coordinates, one can write the 3-metric,
\begin{equation}
\label{IDAxiBrillBH/eqn:sph-coord}
ds^2 = \psi^4 (e^{2q} (dr^2 + r^2 d \theta^2) + r^2 \sin \theta d
\phi^2),
\end{equation}
where $q$ is the Brill ``packet'' which takes some functional form.
Using this ansatz with (\ref{IDAxiBrillBH/eqn:conformal-hamiltonian})
leads to an elliptic equation for $\psi$ which must be solved
numerically. Applying the isometry condition on $\psi$ at a finite
radius, and applying $M/2r$ falloff conditions on $\psi$ at the
outer boundary (the ``Robin'' condition), along with a packet which
obeys the appropriate symmetries (including being invariant under the
isometry operator), will make this solution describe a black hole with
an incident gravitational wave.  The choice of $q=0$ produces the
Schwarzschild solution.   The typical $q$ function used in
axisymmetry, and considered here in the non-rotating case, is 
\begin{equation}
\label{IDAxiBrillBH/eqn:Q}
q = Q_0 \sin^n \theta \left [ \exp\left(\frac{\eta -
      \eta_0^2}{\sigma^2}\right ) + \exp\left(\frac{\eta +
      \eta_0^2}{\sigma^2}\right ) \right ].
\end{equation}
Note regularity along the axis requires that the exponent $n$ must be
even.  Choose a logarithmic radial coordinate $\eta$, which is
related to the
asymptoticlly flat coordinate $r$ by $\eta = \ln (2r/m)$, where $m$ is a
scale parameter.  One can rewrite (\ref{IDAxiBrillBH/eqn:sph-coord}) as
\begin{equation}
ds^2 = \psi(\eta)^4 [ e^{2 q} (d \eta^2 +  d\theta^2) +  \sin^2
\theta d\phi^2].
\end{equation}

In the previous Bernstein work, the above $r$ is transformed to a
logarithmic radial coordinate

\begin{equation}
\label{IDAxiBrillBH/eta-coord}
\eta = \ln{\frac{2r}{m}}.
\end{equation}

The scale parameter $m$ is equal to the mass of the Schwarzschild
black hole, if $q=0$.  In this coordinate, the 3-metric is
\begin{equation}
\label{IDAxiBrillBH/eqn:metric-brill-eta}
ds^2 = \tilde{\psi}^4 (e^{2q} (d\eta^2+d\theta^2)+\sin^2 \theta
d\phi^2),
\end{equation}
and the Schwarzschild solution is 
\begin{equation}
\label{IDAxiBrillBH/eqn:psi}
\tilde{\psi} = \sqrt{2M} \cosh (\frac{\eta}{2}).
\end{equation}
We also change the notation of $\psi$ for the conformal factor is same 
as $\tilde{\psi}$ \cite{Camarda97a}, for the $\eta$ coordinate has the
factor $r^{1/2}$ in the conformal factor.  Clearly $\psi(\eta)$ and
$\psi$ differ by a factor of $\sqrt{r}$.  The Hamiltonian 
constraint is
\begin{equation}
\label{IDAxiBrillBH/eqn:ham}
\frac{\partial^2 \tilde{\psi}}{\partial \eta^2} + \frac{\partial^2
  \tilde{\psi}}{\partial \theta^2} + \cot \theta \frac{\partial
  \tilde{\psi}}{\partial \theta} = - \frac{1}{4} \tilde{\psi}
(\frac{\partial^2 q}{\partial \eta^2} + \frac{\partial^2 q}{\partial
  \theta^2} -1).
\end{equation}

For solving this Hamiltonian constraint numerically.  At first
we substitute
\begin{eqnarray}
\delta \tilde{\psi} & = & \tilde{\psi}+\tilde{\psi}_0 \\
                    & = & \tilde{\psi}-\sqrt{2m} \cosh(\frac{\eta}{2}).
\end{eqnarray}
to the equation~(\ref{IDAxiBrillBH/eqn:ham}), then we can linearize it as
\begin{equation}
\frac{\partial^2 \delta\tilde{\psi}}{\partial \eta^2} + \frac{\partial^2
  \delta\tilde{\psi}}{\partial \theta^2} + \cot \theta \frac{\partial
  \delta\tilde{\psi}}{\partial \theta} = - \frac{1}{4}
(\delta\tilde{\psi} + \tilde{\psi}_0) (\frac{\partial^2 q}{\partial
  \eta^2} + \frac{\partial^2 q}{\partial \theta^2} -1).
\label{IDAxiBrillBH/eqn:ham-linear}
\end{equation}
For the boundary conditions, we use for the inner boundary condition
an isometry condition:
\begin{equation}
\frac{\partial \tilde{\psi}}{\partial \eta}|_{\eta = 0} = 0,
\end{equation}
and outer boundary condition, a Robin condition:
\begin{equation}
(\frac{\partial \tilde{\psi}}{\partial \eta} + \frac{1}{2}
\tilde{\psi})|_{\eta=\eta_{max}} = 0.
\end{equation}

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\section{2-D Grid and Interpolation Parameters}

This thorn solves equation~(\ref{IDAxiBrillBH/eqn:ham-linear}) on a 2-D
$(\eta,\theta)$ grid.  However, Cactus needs a 3-D grid, typically with
Cartesian coordinates.  Therefore, this thorn interpolate $\psi$ and its
$(\eta,\theta)$ derivatives to the Cartesian grid.

The parameters \verb|neta| and \verb|nq| specify the resolution of
this thorn's 2-D grid in $\eta$ and $\theta$ respectively.%%%
\footnote{%%%
	 Internally, this thorn uses ``$q$'' to refer
	 to $\theta$ in Fortran code, with the $q$ function
	 of~$(\protect\ref{IDAxiBrillBH/eqn:Q})$ being hidden
	 in the Mathematica files (and not present in the Fortran
	 code).  Noone seems to know \emph{why} the code does
	 things this way\dots{}  Unfortunately, this renaming
	 has leaked out into the parameter names\dots
	 }%%%
{}  The default values are a reasonable starting point, but you may
need to increase them substantially if you need very high accuracy
(very small constraint violations).

To help judge what resolution may be needed, this thorn has an option
to write out $\psi(\eta)$ and $\psi$ on the 2-D grid to an ASCII data file
where it can be examined and/or plotted.  To do this, set the Boolean
parameter \verb|output_psi2D|, and possibly also the string parameter
\verb|output_psi2D_file_name|.

This thorn uses the standard Cactus \verb|CCTK_InterpLocalUniform()|
local interpolation system for this interpolation.  The interpolation
operator is specified with the \verb|interpolator_name| parameter
(this defaults to \verb|"uniform cartesian"|, the interpolation
operator provided by thorn \textbf{CactusBase/LocalInterp}).

The interpolation order and/or other parameters can be specified
in either of two ways:%%%
\footnote{%%%
	 Notice that, for historical reasons, the
	 interpolation parameter names are a bit
	 inconsistent: \texttt{interpolat\underline{ion}\_order}
	 versus \texttt{interpolat\underline{or}\_name}
	 and \texttt{interpolat\underline{or}\_pars}.
	 }%%%
\begin{itemize}
\item	The integer parameter \verb|interpolation_order| may be
	used directly to specify the interpolation order.
\item	More generally, the string parameter \verb|interpolator_pars|
	may be set to any nonempty string (it defaults to the empty string).
	If this is done, this parameter overrides \verb|interpolation_order|,
	and explicitly specifies a parameter string for the interpolator.
\end{itemize}
Note that the default interpolator parameters specify \emph{linear}
interpolation.  This is rather inaccurate, and (due to aliasing effects
between the 2-D and 3-D grids) will give a fair bit of noise in the
metric components.  You may want to specify a higher-order interpolator
to reduce this noise.

For example, for one test series where I (JT) needed very accurate
initial data (I wanted the initial-data errors to be much less than
the errors from 4th~order finite differencing on the 3-D Cactus grid),
I had to use a resolution of $1000$ in $\eta$ and $2000$ in $\theta$,
together with either 4th~order Lagrange or 3rd~order Hermite interpolation
(provided by thorn \textbf{AEIThorns/AEILocalInterp}) to get sufficient
accuracy.

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\section{Debugging Parameters}

This thorn has options to print very detailed debugging information
about internal quantities at selected grid points.  This is enabled
by setting the integer parameter \verb|debug| to a positive value
(the default is $0$, which means no debugging output).  See
\verb|param.ccl| and the source code \verb|src/IDAxiBrillBH.F| for details.

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\bibliographystyle{prsty}
\begin{thebibliography}{10}
\bibitem{Bernstein93a}
  D. Bernstein, Ph.D thesis University of Illinois Urbana-Champaign,
  (1993)
\bibitem{Brill59}
  D. S. Brill,Ann. Phys.{\bf 7}, 466 (1959)
\bibitem{Camarda97a}
  K. Camarda, Ph.D thesis University of Illinois Urbana-Champaign, (1998)
\end{thebibliography}

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\end{document}