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Diffstat (limited to 'doc/documentation.tex')
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1 files changed, 43 insertions, 36 deletions
diff --git a/doc/documentation.tex b/doc/documentation.tex index 49731f9..a17be7a 100644 --- a/doc/documentation.tex +++ b/doc/documentation.tex @@ -3,13 +3,13 @@ \begin{document} \title{IDBrillData} -\author{Carsten Gundlach} -\date{6 September 1999} +\author{Carsten Gundlach, Gabrielle Allen} +\date{$Date$} \maketitle -\abstract{This thorn creates initial data for Brill wave spacetimes. -It can create both axisymmetric data (in a 3D cartesian grid), as -well as data with an angular dependency.} +\abstract{This thorn creates time symmetric initial data for Brill +wave spacetimes. It can create both axisymmetric data (in a 3D +cartesian grid), as well as data with an angular dependency.} \section{Purpose} @@ -22,40 +22,38 @@ ds^2 = \Psi^4 \left[ e^{2q} \left( d\rho^2 + dz^2 \right) \label{eqn:brillmetric} \end{equation} where $q$ is a free function subject to certain regularity and -fall-off conditions and $\Psi$ is a conformal factor to be solved for. +fall-off conditions, $\rho=\sqrt{x^2+y^2}$ and $\Psi$ is a conformal +factor to be solved for. -The thorn considers several different forms of the function $q$ -depending on certain parameters that will be described below. -Substituting the metric above into the Hamiltonian constraint results -in an elliptic equation for the conformal factor $\Psi$ that can be -solved numerically once the function $q$ has been specified. The -initial data is also assumed to be time-symmetric, so the momentum +Substituting this metric into the Hamiltonian constraint gives an +elliptic equation for the conformal factor $\Psi$ which is then +numerically solved for a given function $q$: +\begin{equation} +\hat{\nabla} \Psi - \frac{\Psi}{8} \hat{R} = 0 +\end{equation} +where the conformal Ricci scalar is found to be +\begin{eqnarray} +\hat{R} = -2 \left(e^{-2q} (\partial^2_z q + \partial^2_\rho q) + +\frac{1}{\rho^2} (3 (\partial_\phi q)^2 + 2 \partial_\phi q)\right) +\end{eqnarray} +Assuming the initial data to be time symmetric means that the momentum constraints are trivially satisfied. -% [[ DPR: The code does not use these two different initial_data -% keywords, but the axisym parameter instead. It should probably use -% the two different initial_data values. I have changed the doc below -% to correspond with the current code: ]] +The thorn considers several different forms of the function $q$ +depending on certain parameters that will be described below. -The thorn is activated by choosing the CactusEinstein/ADMBase parameter -``initial\_data'' to be: -%in one of the following two ways: +Brill initial data is activated by choosing the {\tt CactusEinstein/ADMBase} +parameter {\tt initial\_data} to be {\tt brilldata}. -\begin{itemize} - -\item initial\_data = ``brilldata'': %Axisymmetric -Brill wave initial data -% (but calculated in a cartesian grid!). - -%\item initial\_data = ``brilldata3D'': Brill wave initial data with an -% angular dependency. - -\end{itemize} +In the case of axisymmetry, the Hamiltonian constraint can be written +as an elliptic equation for $\Psi$ with just the flat space Laplacian, +\begin{equation} +\nabla_{flat} \Psi + \frac{\Psi}{4} (\partial_z^2 q + \partial_\rho^2 q) = 0 +\end{equation} +If the initial data is chosen to be {\tt +ADMBase::initial\_data = "brilldata2D"} then this elliptic equation +is solved rather than the equation above. -To choose axisymmetric data, set the parameter {\tt axisym} to true -(the default). Note that the data is computed on a cartesian grid in -any event. {\tt axisym = "no"} adds an angular dependence factor, -which is detailed below. \section{Parameters for the thorn} @@ -128,14 +126,23 @@ The elliptic solver is controlled by the additional parameters: \begin{itemize} -\item solver (KEYWORD): Elliptic solver used to solve the - hamiltonian constraint [sor/petsc/bam] (default ``sor''). +\item {\tt solver} (KEYWORD): Elliptic solver used to solve the + hamiltonian constraint [sor/petsc/bam] (default "sor"). -\item thresh (REAL): Threshold for elliptic solver (default +\item {\tt thresh} (REAL): Threshold for elliptic solver (default 0.00001). \end{itemize} +\section{Notes} + +Thorn {\tt IDBrillData} understands both the {\tt physical} and {\tt +static conformal} {\tt metric\_type}. In the case of a conformal +metric being chosen, the conformal factor is set to $\Psi$. Currently +the derivatives of the conformal factor are not calculated, so that +only {\tt staticconformal::conformal\_storage = "factor"} is +supported. + % Automatically created from the ccl files % Do not worry for now. |