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\documentclass{article}

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%  thorn without the Cactus Flesh download this from the Cactus
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\begin{document}

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

\maketitle

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% Add all definitions used in this documentation here
%   \def\mydef etc
\def\del{\nabla}

\begin{abstract}
Provides macros for common relativity calculations, using the {\bf ADMBase} variables.
\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.

\subsection{Finite Differencing}

By default, the macros use centered 2nd~order finite differencing,
with 3-point finite difference molecules.
That is, when finite differencing the the grid-point indices
${\tt i} \pm 1$, ${\tt j} \pm 1$, and ${\tt k} \pm 1$ must also be valid.

Some of the macros also support centered 4th~order finite differencing;
This is selected with the parameter \verb|spatial_order|.  This may be
set to either~$2$ or~$4$; it defaults to~$2$.  If it's set to~$4$, then
5-point finite difference molecules are used, so the grid-point indices
${\tt i} \pm 2$, ${\tt j} \pm 2$, and ${\tt k} \pm 2$ must also be valid.

At present 4th~order finite differencing is only supported for Fortran code.
(That is, at present the C~versions of the macros simply ignore the
\verb|spatial_order| parameter.)

\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.

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.}

\subsection{Fortran}

If you are using the macros inside a Fortran function then the
{\tt i}, {\tt j} and {\tt k} indices are used directly.  This is
shown in the Fortran example below.

\subsection{C}

If you are using the macros inside a C function then you must 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.  That is, you must define
{\tt di = 1}, {\tt dj = cctk\_lsh[0]} and {\tt dk = cctk\_lsh[0]*cctk\_lsh[1]}.
The C example below should make this clearer.

\section{Examples}

\subsection{Fortran}

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

\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}

\subsection{C}

This function computes the curved-space Laplacian of a scalar field $\phi$,
$\del^i \del_i \phi
	= g^{ij} \partial_{ij} \phi - g^{ij} \Gamma^k_{ij} \partial_k \phi$
assuming that the partial derivatives $\partial_{ij} \phi$ and $\partial_k \phi$
have already been computed:

\newpage	% we'd like the entire example to fit on a single page
\begingroup
\footnotesize
\begin{verbatim}
/*
 * This function computes the curved-space Laplacian of a scalar field,
 * $\del^i \del_i \phi
 *     = g^{ij} \partial_{ij} \phi - g^{ij} \Gamma^k_{ij} \partial_k \phi$
 * at the interior grid points only; it doesn't do anything at all on the
 * boundaries.
 *
 * This function uses the following Cactus grid functions:
 * input:   dx_phi, dy_phi, dz_phi       # 1st derivatives of phi
 *          dxx_phi, dxy_phi, dxz_phi,   # 2nd derivatives of phi
 *                   dyy_phi, dyz_phi,
 *                            dzz_phi
 * output:  Laplacian_phi
 */
void compute_Laplacian(CCTK_ARGUMENTS)
{
DECLARE_CCTK_ARGUMENTS
int i,j,k;

/* contracted Christoffel symbols $\Gamma^k = g^{ij} \Gamma^k_{ij}$ */
CCTK_REAL Gamma_u_x, Gamma_u_y, Gamma_u_z;

/* magic variables for ADMMacros (see the ADMMacros thorn guide for details) */
const int di = 1;
const int dj = cctk_lsh[0];
const int dk = cctk_lsh[0]*cctk_lsh[1];

/* declare the ADMMacros variables for $g^{ij}$ and $\Gamma^k_{ij}$ */
#include "CactusEinstein/ADMMacros/src/macro/UPPERMET_declare.h"
#include "CactusEinstein/ADMMacros/src/macro/CHR2_declare.h"

    for (k = 1 ; k < cctk_lsh[2]-1 ; ++k)
    {
    for (j = 1 ; j < cctk_lsh[1]-1 ; ++j)
    {
    for (i = 1 ; i < cctk_lsh[0]-1 ; ++i)
    {
    const int ijk = CCTK_GFINDEX3D(cctkGH,i,j,k);   /* another magic variable for ADMMacros */

    /* compute the ADMMacros $g^{ij}$ and $\Gamma^k_{ij}$ variables at this grid point */
    #include "CactusEinstein/ADMMacros/src/macro/UPPERMET_guts.h"
    #include "CactusEinstein/ADMMacros/src/macro/CHR2_guts.h"

    /* compute the contracted Christoffel symbols $\Gamma^k = g^{ij} \Gamma^k_{ij}$ */
    Gamma_u_x =
      UPPERMET_UXX*CHR2_XXX + 2.0*UPPERMET_UXY*CHR2_XXY + 2.0*UPPERMET_UXZ*CHR2_XXZ
                            +     UPPERMET_UYY*CHR2_XYY + 2.0*UPPERMET_UYZ*CHR2_XYZ
                                                        +     UPPERMET_UZZ*CHR2_XZZ;
    Gamma_u_y =
      UPPERMET_UXX*CHR2_YXX + 2.0*UPPERMET_UXY*CHR2_YXY + 2.0*UPPERMET_UXZ*CHR2_YXZ
                            +     UPPERMET_UYY*CHR2_YYY + 2.0*UPPERMET_UYZ*CHR2_YYZ
                                                        +     UPPERMET_UZZ*CHR2_YZZ;
    Gamma_u_z =
      UPPERMET_UXX*CHR2_ZXX + 2.0*UPPERMET_UXY*CHR2_ZXY + 2.0*UPPERMET_UXZ*CHR2_ZXZ
                            +     UPPERMET_UYY*CHR2_ZYY + 2.0*UPPERMET_UYZ*CHR2_ZYZ
                                                        +     UPPERMET_UZZ*CHR2_ZZZ;

    /* compute the Laplacian */
    Laplacian_phi[ijk] =
      UPPERMET_UXX*dxx_phi[ijk] + 2.0*UPPERMET_UXY*dxy_phi[ijk] + 2.0*UPPERMET_UXZ*dxz_phi[ijk]
                                +     UPPERMET_UYY*dyy_phi[ijk] + 2.0*UPPERMET_UYZ*dyz_phi[ijk]
                                                                +     UPPERMET_UZZ*dzz_phi[ijk]
      -  Gamma_u_x*dx_phi[ijk]  -        Gamma_u_y*dy_phi[ijk]  -        Gamma_u_z*dz_phi[ijk];
    }
    }
    }

#include "CactusEinstein/ADMMacros/src/macro/UPPERMET_undefine.h"
#include "CactusEinstein/ADMMacros/src/macro/CHR2_undefine.h"
}
\end{verbatim}
\endgroup

\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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