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\begin{document}

\title{ADMConstraints}
\author{Gabrielle Allen}
\date{$ $Date$ $}

\maketitle

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\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}

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