\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{ADMMacros}
\author{Tom Goodale}
\date{$ $Date$ $}

\maketitle

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\begin{abstract}
Provides macros for other ADM based thorns to use
\end{abstract}

\section{Purpose}

This thorn provides various macros which can be used to calculate
quantities, such as the Christoffel Symbol or Riemann Tensor
components, using the basic variables of thorn {\tt ADMBase} (and {\tt
  StaticConformal} if required).  The macros can be used in both
Fortran and C code. The macros work pointwise to calculate quantities
at specific grid points, which are always labelled by {\tt i}, {\tt j}
and {\tt k}. They are written in such a way that needed quantities
which are already calculated are reused.

\section{Using ADM Macros}

Each macro described in Section~\ref{admmacros:macros} is implemented using three include 
files. 

\begin{itemize}

\item {\tt <MACRONAME>\_declare.h} sets up the declarations for the internal macro 
        variables. All the internal (hidden) variables have names beginning with 
        the macro name. This file should be included in the declarations section
        of your routine.

\item {\tt <MACRONAME>\_guts.h} is the actual included source code which will calculate
       the quantities.

\item {\tt <MACRONAME>\_undefine.h} resets the macros. This file {\bf must be included}
        at the end of every loop using macros. Without an undefine file, a second loop
        using macros will assume that quantities have already been calculated.

\end{itemize}

To use the macros, first make sure that you really want to use the
macro pointwise, and that you have already set the indices $i$, $j$,
and $k$ to identify the correct grid point. If you are using the
macros inside a Fortran function then the {\tt i}, {\tt j} and {\tt k}
indices are used directly. If you are using the macros inside a C
function then you should define the index {\tt ijk}, which can be
found from {\tt i}, {\tt j} and {\tt k} using the macro {\tt
  CCTK\_GFINDEX3D(cctkGH,i,j,k)}, and also the offsets {\tt di}, {\tt
  dj} and {\tt dk}, so that the point $(i-1,j,k)$ is the same as {\tt
  ijk - di} and so on. Typically this would mean that {\tt dk = 1},
{\tt dj = cctk\_lsh[2]} and {\tt di = cctk\_lsh[1]*cctk\_lsh[2]}, where
C-style numbering is used.

Find the name of the macro from the table in
Section~\ref{admmacros:macros} and put the include files in the
correct place following the instructions above. Note that all ADMMacro
include files are in the directory {\tt
  CactusEinstein/ADMMacros/src/macros}, so this means adding lines
such as

{\tt
\begin{verbatim}
#include "CactusEinstein/ADMacros/src/macro/<MACRONAME>_<TYPE>.h"
\end{verbatim}
}

Each variable that the macro calculates is listed in the table of
Section~\ref{admmacros:macros}.  Note that these variable names are
themselves macros and are case sensitive. {\bf Always use the macro
variables on the right hand sides of equations, never redefine them
yourself, since they may be used in later (hidden) calculations.}

\section{Example}

This example comes from thorn {\tt CactusEinstein/Maximal} and uses the $trK$ macro to 
calculate the trace of the extrinsic curvature. 

{\tt
\begin{verbatim}
c     Declarations for macros.
#include "CactusEinstein/ADMMacros/src/macro/TRK_declare.h"

<CUT> 

c     Add the shift term: N = B^i D_i(trK).
      if ((maxshift).and.(shift_state.eq.1)) then
         do k=1,nz
            do j=1,ny
               do i=1,nx
#include "CactusEinstein/ADMMacros/src/macro/TRK_guts.h"
                  K_temp(i,j,k) = TRK_TRK
                end do
             end do
         end do
#include "CactusEinstein/ADMMacros/src/macro/TRK_undefine.h"
\end{verbatim}
}


\section{Macros}
\label{admmacros:macros}
Macros exist for the following quantities

\begin{tabular}{p{5cm}p{5cm}p{5cm}}
{\bf Calculates} & {\bf Macro Name} & {\bf Sets variables} \\
All first spatial derivatives of lapse, $\alpha_{,i}$: & DA & DA\_DXDA, DA\_DYDA, DA\_DZDA\\
All second spatial derivatives of lapse, $\alpha_{,ij}$: & DDA & DDA\_DXXDA, DDA\_DXYDA, DDA\_DXZDA, DDA\_DYYDA, DDA\_DYZDA, DDA\_DZZDA\\
All second covariant spatial derivatives of lapse, $\alpha_{;ij}$: & CDCDA &\\
All first spatial derivatives of shift, $\beta^{i}_{\;\;j}$: & DB &\\
All first covariant derivatives of the extrinsic curvature, $K_{ij;kl}$ & CDK &\\
First covariant derivatives of the extrinsic curvature, $K_{ij;x}$, $K_{ij;y}$, $K_{ij;z}$ & CDXCDK, CDYCDK, CDZCDK &\\
Determinant of 3-metric: & DETG &\\
Upper 3-metric, $g{ij}$:& UPPERMET &\\
Trace of extrinsic curvature $trK$: & TRK &\\
Trace of stress energy tensor: & TRT &\\
Hamiltonian constraint: & HAMADM & \\
Partial derivatives of extrinsic curvature, $K_{ij,x}$, $K_{ij,y}$, $K_{ij,z}$: & DXDK, DYDK, DZDK &\\
First partial derivatives of 3-metric, $g_{ij,x}$, $g_{ij,y}$, $g_{ij,z}$: & DXDG, DYDG, DZDG & \\
All first partial derivatives of 3-metric, $g_{ij,k}$: & DG &\\
First covariant derivatives of 3-metric, $g_{ij;x}$, $g_{ij;y}$, $g_{ij;z}$: & DXDCG, DYDCG, DZDCG &\\
Second partial derivatives of 3-metric, $g_{ij,xx}$, $g_{ij,xy}$, $g_{ij,xz}$: & DXXDG, DXYDG, DXZDG, DYYDG, DYZDG, DZZDG& \\
All second partial derivative of 3-metric, $g_{ij,lm}$ & DDG &\\
Ricci tensor $R_{ij}$: & RICCI &\\
Trace of Ricci tensor $R$: & TRRICCI &\\
Christoffel symbols of first kind: $\Gamma_{cab}$ & CHR1&\\
Christoffel symbols of second kind $\Gamma^{c}_{\;\;ab}$: & CHR2& \\
Momentum constraints & MOMX, MOMY, MOMZ&\\
Source term in evolution equation for conformal metric, $\tilde{g}_{ij,t}$: & DCGDT &\\

\end{tabular}

\section{Definitions}

\begin{equation}
\Gamma_{cab} = \frac{1}{2}\left(g_{ac,b} + g_{bc,a} - g_{ab,c}\right)
\end{equation}

\begin{equation}
\Gamma^{c}_{\;\;ab} = g^{cd}\Gamma_{dab} = \frac{1}{2} g^{cd} \left(g_{ad,b} + g_{bd,a} - g_{ab,d}\right)
\end{equation}

%\section{Comments}

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