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Author, {\em The Title of the Book, Journal, or periodical}, 1 (1999), % 1--16. {\tt http://www.nowhere.com/}} % % *======================================================================* % If you are using CVS use this line to give version information % $Header$ \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{ADMMacros} \author{Tom Goodale} \date{$ $Date$ $} \maketitle % Do not delete next line % START CACTUS THORNGUIDE % 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. \section{Using ADM Macros} Each macro described in Section~\ref{admmacros:macros} is implemented using three include files. \begin{itemize} \item {\tt \_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 \_guts.h} is the actual included source code which will calculate the quantities. \item {\tt \_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/_.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" 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 $\phi$, $\del^i \del_i \phi = g^{ij} \partial_{ij} \phi - g^{ij} \Gamma^k_{ij} \partial_k \phi$ using centered 2nd order finite differencing. \newpage \begingroup \footnotesize \begin{verbatim} /* * This function computes the curved-space Laplacian of a scalar field, * $$ * \del^i \del_i \phi * = g^{ij} \partial_{ij} \phi - g^{ij} \Gamma^k_{ij} \partial_k \phi * $$ * 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 = 0 ; k < cctk_lsh[2] ; ++k) { for (j = 0 ; j < cctk_lsh[1] ; ++j) { for (i = 0 ; i < cctk_lsh[0] ; ++i) { const int ijk = CCTK_GFINDEX3D(cctkGH,i,j,k); /* another magic variable for ADMMacros */ /* compute the ADMMacros $g^{ij}$ and $\Gamma^k_{ij}$ */ #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} % Do not delete next line % END CACTUS THORNGUIDE \end{document}