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\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/latex/cactus}
\begin{document}
\title{ADMConstraints}
\author{Gabrielle Allen}
\date{$ $Date$ $}
\maketitle
% Do not delete next line
% START CACTUS THORNGUIDE
\begin{abstract}
Calculate the values of the Hamiltonian and Momentum Constraints on the computational grid for output.
\end{abstract}
\section{Physics}
This analysis thorn calculates the values of the Hamiltonian and
momentum constraint equations on the computational grid during a spacetime
evolution. The Hamiltonian Constraint (grid function: ham) is given by
\begin{equation}
H = R - K^{i}_{\;\;j} K^{j}_{\;\;i} + trK^2 - 16 \pi \rho
\end{equation}
The Momentum Constraints (grid functions momx, momy, momz) are given by
\begin{equation}
M_i = \nabla_j K_i^{\;\;j} - \nabla_i trK - 8 \pi S_i
\end{equation}
where
\begin{equation}
S_i = -g_{ia} n_b T^{ab} = - g_i^a n^b T_{ab} = -\frac{1}{\alpha} (T_{i0} - \beta^j T_{ij})
\end{equation}
\section{Comments}
\begin{itemize}
\item Symmetry boundary conditions ({\tt CactusBase/CartGrid3D}) are implemented.
\item Default behaviour is to apply so called flat boundary conditions to the calculated values of the constraint equations.
\item Excision ({\tt Excision/LegoExcision}) is included.
\item The 3-metric is only included through the Macros in {\tt CactusEinstein/Einstein} and as such the physical metric is used.
\item Matter is included using the {\tt CalcTmuNu} mechanism.
\item The constraint calculation is default performed at the ANALYSIS time bin, if the constraint values are needed at every iteration, the parameter {\tt constraints\_persist} should be used to schedule the calculation instead at POSTSTEP.
\end{itemize}
% Do not delete next line
% END CACTUS THORNGUIDE
\end{document}
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