C

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

/*
 * 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;

/* grid-function strides for ADMMacros */
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);   /* grid-function subscripting index for ADMMacros */
                                                    /* (must be assigned inside the i,j,k loops) */

    /* compute the ADMMacros $g^{ij}$ and $\Gamma^k_{ij}$ variables at the (i,j,k) 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"
}