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diff --git a/doc/documentation.tex b/doc/documentation.tex index a17be7a..2703ebb 100644 --- a/doc/documentation.tex +++ b/doc/documentation.tex @@ -13,9 +13,9 @@ cartesian grid), as well as data with an angular dependency.} \section{Purpose} -The purpose of this thorn is to create initial data for a Brill wave -spacetime. It does so by starting from a three--metric of the form -originally considered by Brill +The purpose of this thorn is to create (time symmetric) initial data +for a Brill wave spacetime. It does so by starting from a +three--metric of the form originally considered by Brill \begin{equation} ds^2 = \Psi^4 \left[ e^{2q} \left( d\rho^2 + dz^2 \right) + \rho^2 d\phi^2 \right] =\Psi^4 \hat{ds}^{2}, @@ -25,7 +25,29 @@ where $q$ is a free function subject to certain regularity and fall-off conditions, $\rho=\sqrt{x^2+y^2}$ and $\Psi$ is a conformal factor to be solved for. -Substituting this metric into the Hamiltonian constraint gives an +Thorn {\tt IDBrillData} provides three choices for the $q$ function: +an exponential form, ({\tt IDBrillData::q\_function = "exp"}) +\begin{equation} +q = a \; \frac{\rho^{2+b}}{r^2} \left( \frac{z}{\sigma_z} \right)^2 +e^{-(\rho - \rho_0^2)} \left[ 1 + d \frac{\rho^m}{1 + e \rho^m} +\cos^2 \left( n \phi + \phi_0 \right) \right] +\end{equation} +a generalized form of the $q$ function first written down by Eppley +({\tt IDBrillData::q\_function = "eppley"}) +\begin{equation} +q = a \left( \frac{\rho}{\sigma_\rho} \right)^b \frac{1}{1 + \left[ +\left( r^2 - r_0^2 \right) / \sigma_r^2 \right]^{c/2}}\left[ 1 + d \frac{\rho^m}{1 + e \rho^m} +\cos^2 \left( n \phi + \phi_0 \right) \right] +\end{equation} +and the (default) Gundlach $q$ function which includes the Holz form +({\tt IDBrillData::q\_function = "gundlach"}) +\begin{equation} +q = a \left( \frac{\rho}{\sigma_\rho} \right)^b e^{-\left[ +\left( r^2 - r_0^2 \right) / \sigma_r^2 \right]^{c/2}} \left[ 1 + d \frac{\rho^m}{1 + e \rho^m} +\cos^2 \left( n \phi + \phi_0 \right) \right] +\end{equation} + +Substituting the 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} @@ -39,14 +61,9 @@ where the conformal Ricci scalar is found to be Assuming the initial data to be time symmetric means that the momentum constraints are trivially satisfied. -The thorn considers several different forms of the function $q$ -depending on certain parameters that will be described below. - -Brill initial data is activated by choosing the {\tt CactusEinstein/ADMBase} -parameter {\tt initial\_data} to be {\tt brilldata}. - -In the case of axisymmetry, the Hamiltonian constraint can be written -as an elliptic equation for $\Psi$ with just the flat space Laplacian, +In the case of axisymmetry (that is $d=0$ in the above expressions for +$q$), 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} @@ -55,82 +72,60 @@ ADMBase::initial\_data = "brilldata2D"} then this elliptic equation is solved rather than the equation above. -\section{Parameters for the thorn} +\section{Generating Initial Data with IDBrillData} -The thorn is controlled by the following parameters: +Brill initial data is activated by choosing the {\tt CactusEinstein/ADMBase} +parameter {\tt initial\_data} to be {\tt brilldata}, or for the case of +axisymmetry {\tt brilldata2D} can also be used. -\begin{itemize} +The parameter {\tt IDBrillData::q\_function} chooses the form of the +$q$ function to be used, defaulting to the Gundlach expression. -\item brill\_q (INT): Form of the function $q$ [0,1,2] (default 2): +Additional {\tt IDBrillData} parameters for each form of $q$ fix the +remaining freedom: \begin{itemize} -\item brill\_q = 0: -\[ -q = a \; \frac{\rho^{2+b}}{r^2} \left( \frac{z}{\sigma_z} \right)^2 -e^{-(\rho - \rho_0^2)} -\] +\item Exponential $q$: {\tt IDBrillData::q\_function = "exp"} -\item brill\_q = 1: -\[ -q = a \left( \frac{\rho}{\sigma_\rho} \right)^b \frac{1}{1 + \left[ -\left( r^2 - r_0^2 \right) / \sigma_r^2 \right]^{c/2}} -\] +$(a, b,\sigma_z,\rho_0)=$ ({\tt exp\_a, exp\_b, exp\_sigmaz,exp\_rho0}) -\item brill\_q = 2: -\[ -q = a \left( \frac{\rho}{\sigma_\rho} \right)^b e^{-\left[ -\left( r^2 - r_0^2 \right) / \sigma_r^2 \right]^{c/2}} -\] +\item Eppley $q$: {\tt IDBrillData::q\_function = "eppley"} -\item If one specifies 3D data (see above), the function $q$ is multiplied -by an additional factor with an angular dependency: -\[ -q \rightarrow q \left[ 1 + d \frac{\rho^m}{1 + e \rho^m} -\cos^2 \left( n \phi + \phi_0 \right) \right] -\] +$(a, b,\sigma_\rho, r_0,\sigma_r,c)=$ ({\tt eppley\_a, eppley\_b, eppley\_sigmarho, eppley\_r0, eppley\_sigmar, eppley\_c}) -\end{itemize} - -\item brill\_a (REAL): Amplitude (default 0.0). - -\item brill\_b (REAL): $b$ in above expressions (default 2.0). - -\item brill\_c (REAL): $c$ in above expressions (default 2.0). - -\item brill\_d (REAL): $d$ in above expressions (default 0.0). +\item Gundlach $q$: {\tt IDBrillData::q\_function = "gundlach"} -\item brill\_e (REAL): $e$ in above expressions (default 1.0). +$(a, b,\sigma_\rho, r_0,\sigma_r,c)=$ ({\tt gundlach\_a, gundlach\_b, gundlach\_sigmarho, gundlach\_r0, gundlach\_sigmar, gundlach\_c}) -\item brill\_m (REAL): $m$ in above expressions (default 2.0). +\item Non-axisymmetric part for each choice of $q$ -\item brill\_n (REAL): $n$ in above expressions (default 2.0). +$(d, m, e, n, \phi0)=$ ({\tt brill3d\_d, brill3d\_m, brill3d\_e, brill3d\_n, brill3d\_phi0}) -\item brill\_r0 (REAL): $r_0$ in above expressions (default 0.0). - -\item brill\_rho0 (REAL): $\rho_0$ in above expressions - (default 0.0). - -\item brill\_phi0 (REAL): $\phi_0$ in above expressions - (default 0.0). - -\item brill\_sr (REAL): $\sigma_r$ in above expressions - (default 1.0). - -\item brill\_srho (REAL): $\sigma_\rho$ in above - expressions (default 1.0). - \end{itemize} -The elliptic solver is controlled by the additional parameters: +Note that the default $q$ expression is +$$ +q = {\tt gundlach\_a} \quad \rho^2 e^{-r^2} +$$ + +{\tt IDBrillData} can use the elliptic solvers (type LinMetric) +provided by {\tt CactusEinstein/EllSOR}, {\tt AEIThorns/BAM\_Elliptic} +or {\tt CactusElliptic/EllPETSc} to solve the equation resulting from +the Hamiltonian constraint. +In all cases the parameter {\tt thresh} sets the threshold for the elliptic +solve. The choice of elliptic solver is made +through the parameter {\tt brill\_solver}: \begin{itemize} -\item {\tt solver} (KEYWORD): Elliptic solver used to solve the - hamiltonian constraint [sor/petsc/bam] (default "sor"). +\item {\tt sor}: Understands the Robin boundary condition, additional +parameters control the maximum number of iterations ({\tt sor\_maxit}). -\item {\tt thresh} (REAL): Threshold for elliptic solver (default - 0.00001). +\item {\tt bam}: {\tt BAM\_Elliptic} does not properly implement the +elliptic infrastructure of {\tt EllBase}, and the {\tt BAM\_Elliptic} +parameter to use the Robin boundary condition must be set independently +of {tt IDBrillWave::brill\_bound}. \end{itemize} @@ -143,6 +138,29 @@ the derivatives of the conformal factor are not calculated, so that only {\tt staticconformal::conformal\_storage = "factor"} is supported. +\section{References} + +\subsection{Specification of Brill Waves} + +\begin{enumerate} + +\item Dieter Brill, {\bf Ann. Phys.}, 7, 466, 1959. + +\item Ken Eppley, {\bf Sources of Gravitational Radiation}, edited by L. Smarr (Cambridge University Press, +Cambridge, England, 1979), p. 275. + +\end{enumerate} + +\subsection{Numerical Evolutions of Brill Waves} + +\begin{enumerate} + +\item {\it Gravitational Collapse of Gravitational Waves in 3D Numerical Relativity}, + Miguel Alcubierre, Gabrielle Allen, Bernd Bruegmann, Gerd Lanfermann, Edward Seidel, Wai-Mo Suen, Malcolm Tobias, +{\bf Phys. Rev. D61}, 041501, 2000. + +\end{enumerate} + % Automatically created from the ccl files % Do not worry for now. |