binary_helical.C

/*
 * Methods of Bin_star::helical
 *
 * (see file star.h for documentation)
 *
 */

/*
 *   Copyright (c) 2006 Francois Limousin
 *
 *   This file is part of LORENE.
 *
 *   LORENE is free software; you can redistribute it and/or modify
 *   it under the terms of the GNU General Public License as published by
 *   the Free Software Foundation; either version 2 of the License, or
 *   (at your option) any later version.
 *
 *   LORENE is distributed in the hope that it will be useful,
 *   but WITHOUT ANY WARRANTY; without even the implied warranty of
 *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *   GNU General Public License for more details.
 *
 *   You should have received a copy of the GNU General Public License
 *   along with LORENE; if not, write to the Free Software
 *   Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 *
 */

char binary_helical_C[] = "$Header: /cvsroot/Lorene/C++/Source/Binary/binary_helical.C,v 1.4 2008/08/19 06:41:59 j_novak Exp $" ;

/*
 * $Id: binary_helical.C,v 1.4 2008/08/19 06:41:59 j_novak Exp $
 * $Log: binary_helical.C,v $
 * Revision 1.4  2008/08/19 06:41:59  j_novak
 * Minor modifications to avoid warnings with gcc 4.3. Most of them concern
 * cast-type operations, and constant strings that must be defined as const char*
 *
 * Revision 1.3  2006/08/01 14:26:50  f_limousin
 * Small changes
 *
 * Revision 1.2  2006/06/05 17:05:57  f_limousin
 * *** empty log message ***
 *
 * Revision 1.1  2006/04/11 14:25:15  f_limousin
 * New version of the code : improvement of the computation of some
 * critical sources, estimation of the dirac gauge, helical symmetry...
 *

 *
 * $Header: /cvsroot/Lorene/C++/Source/Binary/binary_helical.C,v 1.4 2008/08/19 06:41:59 j_novak Exp $ *
 */


// C headers
#include <math.h>

// Lorene headers
#include "cmp.h"
#include "tenseur.h"
#include "metrique.h"
#include "binary.h"
#include "param.h"
#include "graphique.h"
#include "utilitaires.h"
#include "tensor.h"
#include "nbr_spx.h"
#include "unites.h"

void Binary::helical(){

    // Fundamental constants and units
    // -------------------------------
    using namespace Unites ;

 
    Sym_tensor lie_aij_1 (star1.mp, CON, star1.mp.get_bvect_cart()) ;
    Sym_tensor lie_aij_2 (star2.mp, CON, star2.mp.get_bvect_cart()) ;

    Scalar lie_K_1 (star1.mp) ;
    Scalar lie_K_2 (star2.mp) ;

    for (int ll=1; ll<=2; ll++) {
    
    Star_bin star_i (*et[ll-1]) ;
    
    Map& mp = star_i.mp ;
    const Mg3d* mg = mp.get_mg() ; 
    int nz = mg->get_nzone() ;      // total number of domains
    
    Metric_flat flat = star_i.flat ;
    Metric gtilde = star_i.gtilde ;
    Scalar nn = star_i.nn ;
    Scalar psi4 = star_i.psi4 ;

    // -------------------------------
    // AUXILIARY QUANTITIES
    // -------------------------------

    // Derivatives of N and logN
    //--------------------------

    const Vector dcov_logn_auto = star_i.logn_auto.derive_cov(flat) ;
    
    Tensor dcovdcov_logn_auto = (star_i.logn_auto.derive_cov(flat))
                                      .derive_cov(flat) ;
    dcovdcov_logn_auto.inc_dzpuis() ;

    // Derivatives of lnq, phi and Q
    //-------------------------------

    const Scalar phi (0.5 * (star_i.lnq - star_i.logn)) ;
    const Scalar phi_auto (0.5 * (star_i.lnq_auto - star_i.logn_auto)) ;

    const Vector dcov_phi_auto = phi_auto.derive_cov(flat) ;
    
    const Vector dcov_lnq = 2*star_i.dcov_phi + star_i.dcov_logn ;
    const Vector dcon_lnq = 2*star_i.dcon_phi + star_i.dcon_logn ;
    const Vector dcov_lnq_auto = star_i.lnq_auto.derive_cov(flat) ;
        Tensor dcovdcov_lnq_auto = dcov_lnq_auto.derive_cov(flat) ;
    dcovdcov_lnq_auto.inc_dzpuis() ;

    Scalar qq = exp(star_i.lnq) ;
    qq.std_spectral_base() ;
    const Vector& dcov_qq = qq.derive_cov(flat) ;

    Tensor dcovdcov_beta_auto = star_i.beta_auto.derive_cov(flat)
      .derive_cov(flat) ;
    dcovdcov_beta_auto.inc_dzpuis() ;


    // Derivatives of hij, gtilde... 
    //------------------------------

    Scalar psi2 (pow(star_i.psi4, 0.5)) ;
    psi2.std_spectral_base() ;

    const Tensor& dcov_hij = star_i.hij.derive_cov(flat) ;
    const Tensor& dcon_hij = star_i.hij.derive_con(flat) ;
    const Tensor& dcov_hij_auto = star_i.hij_auto.derive_cov(flat) ;

    const Sym_tensor& gtilde_cov = star_i.gtilde.cov() ;
    const Sym_tensor& gtilde_con = star_i.gtilde.con() ;
    const Tensor& dcov_gtilde = gtilde_cov.derive_cov(flat) ;

    // H^i and its derivatives ( = O in Dirac gauge)
    // ---------------------------------------------

    double lambda_dirac = 0. ;

    const Vector hdirac = lambda_dirac * star_i.hij.divergence(flat) ;
    const Vector hdirac_auto = lambda_dirac * 
        star_i.hij_auto.divergence(flat) ;

    Tensor dcov_hdirac = lambda_dirac * hdirac.derive_cov(flat) ;
    dcov_hdirac.inc_dzpuis() ;
    Tensor dcov_hdirac_auto = lambda_dirac * hdirac_auto.derive_cov(flat) ;
    dcov_hdirac_auto.inc_dzpuis() ;
    Tensor dcon_hdirac_auto = lambda_dirac * hdirac_auto.derive_con(flat) ;
    dcon_hdirac_auto.inc_dzpuis() ;


    // Function exp(-(r-r_0)^2/sigma^2)
    // --------------------------------
    
    double r0 = mp.val_r(nz-2, 1, 0, 0) ;
    double sigma = 1.*r0 ;
    double om = omega ;
    
    Scalar rr (mp) ;
    rr = mp.r ;
    
    Scalar ff (mp) ;
    ff = exp( -(rr - r0)*(rr - r0)/sigma/sigma ) ;
    for (int ii=0; ii<nz-1; ii++)
        ff.set_domain(ii) = 1. ;
    ff.set_outer_boundary(nz-1, 0) ;
    ff.std_spectral_base() ;
    

    // omdsdp
    // ---------------------------------

    Vector omdsdp (mp, CON, mp.get_bvect_cart()) ;
    Scalar yya (mp) ;
    yya = mp.ya ;
    Scalar xxa (mp) ;
    xxa = mp.xa ;
    Scalar zza (mp) ;
    zza = mp.za ;

    if (fabs(mp.get_rot_phi()) < 1e-10){ 
        omdsdp.set(1) = - om * yya * ff ;
        omdsdp.set(2) = om * xxa * ff ;
        omdsdp.set(3).annule_hard() ;
    }
    else{
        omdsdp.set(1) = om * yya * ff ;
        omdsdp.set(2) = - om * xxa * ff ;
        omdsdp.set(3).annule_hard() ;
    }
 
    omdsdp.set(1).set_outer_boundary(nz-1, 0) ;
    omdsdp.set(2).set_outer_boundary(nz-1, 0) ;     
    omdsdp.std_spectral_base() ;


    // Computation of helical A^{ij}
    // ------------------------------

    const Tensor& dbeta = star_i.beta_auto.derive_con(gtilde) ;
    Scalar div_beta = star_i.beta_auto.divergence(gtilde) ;
    
    Sym_tensor tkij_a (star_i.tkij_auto) ;
    for (int i=1; i<=3; i++) 
        for (int j=1; j<=i; j++) {
        tkij_a.set(i, j) = dbeta(i, j) + dbeta(j, i) - 
            double(2) /double(3) * div_beta * (gtilde.con())(i,j) ; 
        }
    
    tkij_a = tkij_a - star_i.hij_auto.derive_lie(omdsdp) ;  
    tkij_a = 0.5 * tkij_a / nn ;   

    Sym_tensor tkij_auto_cov = tkij_a.up_down(gtilde) ;
    
    Scalar a2_auto = contract(tkij_auto_cov, 0, 1, tkij_a, 0, 1,true) ; 

    // COMP 
    const Tensor& dbeta_comp = star_i.beta_comp.derive_con(gtilde) ;
        Scalar divbeta_comp = star_i.beta_comp.divergence(gtilde) ;
    
    Sym_tensor tkij_c (star_i.tkij_comp) ;
    for (int i=1; i<=3; i++) 
        for (int j=i; j<=3; j++) {      
        tkij_c.set(i, j) = dbeta_comp(i, j) + dbeta_comp(j, i) - 
            double(2) /double(3) * divbeta_comp * (gtilde.con())(i,j) ; 
        }
    
    tkij_c = tkij_c - star_i.hij_comp.derive_lie(omdsdp) ;  
    tkij_c = 0.5 * tkij_c / nn ;
    
    Scalar a2_comp = contract(tkij_auto_cov, 0, 1, tkij_c, 0, 1,true) ; 


//  tkij_a = star_i.tkij_auto ;
//  tkij_c = star_i.tkij_comp ;


    // Sources for N
    // ---------------

    Scalar source1(mp) ;
    Scalar source2(mp) ;
    Scalar source3(mp) ;
    Scalar source4(mp) ;
    Scalar source_tot(mp) ;

    source1 = qpig * star_i.psi4 % (star_i.ener_euler + star_i.s_euler) ; 

    source2 = star_i.psi4 % (a2_auto + a2_comp) ;

    source3 = - contract(dcov_logn_auto, 0, star_i.dcon_logn, 0, true) 
        - 2. * contract(contract(gtilde_con, 0, star_i.dcov_phi, 0), 
                0, dcov_logn_auto, 0, true) ;
            
    source4 = - contract(star_i.hij, 0, 1, dcovdcov_logn_auto + 
                 dcov_logn_auto*star_i.dcov_logn, 0, 1) ;

    source_tot = source1 + source2 + source3 + source4 ;
    
    if (ll == 1) {
    lie_K_1 = nn / star_i.psi4 * (source_tot - star_i.logn_auto
                       .laplacian()) ;
    lie_K_1.dec_dzpuis(4) ;
    }
    if (ll == 2) {
    lie_K_2 = nn / star_i.psi4 * (source_tot - star_i.logn_auto
                       .laplacian()) ;
    lie_K_2.dec_dzpuis(4) ;
    }


    // Sources for hij
    // --------------

    Scalar source_tot_hij(mp) ;
    Tensor source_Sij(mp, 2, CON, mp.get_bvect_cart()) ;
    Tensor source_Rij(mp, 2, CON, mp.get_bvect_cart()) ;
    Tensor tens_temp(mp, 2, CON, mp.get_bvect_cart()) ;
    
    Tensor source_1 (mp, 2, CON, mp.get_bvect_cart()) ;
    Tensor source_2 (mp, 2, CON, mp.get_bvect_cart()) ;
    Tensor source_3a (mp, 2, CON, mp.get_bvect_cart()) ;
    Tensor source_3b (mp, 2, CON, mp.get_bvect_cart()) ;
    Tensor source_4 (mp, 2, CON, mp.get_bvect_cart()) ;
    Tensor source_5 (mp, 2, CON, mp.get_bvect_cart()) ;
    Tensor source_6 (mp, 2, CON, mp.get_bvect_cart()) ;
    

    source_1 = contract(dcon_hij, 1, dcov_lnq_auto, 0) ;
    
    source_2 = - contract(dcon_hij, 2, dcov_lnq_auto, 0) 
        - 2./3. * contract(hdirac, 0, dcov_lnq_auto, 0) * flat.con() ;
        
    // Lie derivative of A^{ij}
    // --------------------------

    Scalar decouple_logn = (star_i.logn_auto - 1.e-8)/
        (star_i.logn - 2.e-8) ;

    // Construction of Omega d/dphi
    // ----------------------------
        
    // Construction of D_k \Phi^i
    Itbl type (2) ;
    type.set(0) = CON ;
    type.set(1) = COV ;

    Tensor dcov_omdsdphi (mp, 2, type, mp.get_bvect_cart()) ;
    dcov_omdsdphi.set(1,1) = 0. ;
    dcov_omdsdphi.set(2,1) = om * ff ;
    dcov_omdsdphi.set(3,1) = 0. ;
    dcov_omdsdphi.set(2,2) = 0. ;
    dcov_omdsdphi.set(3,2) = 0. ;
    dcov_omdsdphi.set(3,3) = 0. ;
    dcov_omdsdphi.set(1,2) = -om *ff ;
    dcov_omdsdphi.set(1,3) = 0. ;
    dcov_omdsdphi.set(2,3) = 0. ;
    dcov_omdsdphi.std_spectral_base() ;

    source_3a = contract(tkij_a, 0, dcov_omdsdphi, 1) ;
    source_3a.inc_dzpuis() ;

    // Source 3b
    // ------------

    source_3b = - contract(omdsdp, 0, tkij_a.derive_cov(flat), 2) ; 

 
    // Source 4
    // ---------

    source_4 = - tkij_a.derive_lie(star_i.beta) ;
    source_4.inc_dzpuis() ;
    source_4 += - 2./3. * star_i.beta.divergence(flat) * tkij_a ;

    source_5 = dcon_hdirac_auto ;
        
    source_6 = - 2./3. * hdirac_auto.divergence(flat) * flat.con() ;
    source_6.inc_dzpuis() ;

    // Source terms for Sij
    //---------------------
        
    source_Sij = 8. * nn / psi4 * phi_auto.derive_con(gtilde) * 
        contract(gtilde_con, 0, star_i.dcov_phi, 0) ;

    source_Sij += 4. / psi4 * phi_auto.derive_con(gtilde) * 
        nn * contract(gtilde_con, 0, star_i.dcov_logn, 0) +
        4. / psi4 * nn * contract(gtilde_con, 0, star_i.dcov_logn, 0) *
        phi_auto.derive_con(gtilde) ;

    source_Sij += - nn / (3.*psi4) * gtilde_con * 
        ( 0.25 * contract(gtilde_con, 0, 1, contract(dcov_hij_auto, 0, 1,
                           dcov_gtilde, 0, 1), 0, 1)
          - 0.5 * contract(gtilde_con, 0, 1, contract(dcov_hij_auto, 0, 1,
                           dcov_gtilde, 0, 2), 0, 1)) ;

    source_Sij += - 8.*nn / (3.*psi4) * gtilde_con * 
     contract(dcov_phi_auto, 0, contract(gtilde_con, 0, star_i.dcov_phi, 0), 0) ;

    tens_temp = nn / (3.*psi4) * hdirac.divergence(flat)*star_i.hij_auto ;
    tens_temp.inc_dzpuis() ;

    source_Sij += tens_temp ;
       
    source_Sij += - 8./(3.*psi4) * contract(dcov_phi_auto, 0,
       nn*contract(gtilde_con, 0, star_i.dcov_logn, 0), 0) * gtilde_con ;

    source_Sij += 2.*nn* contract(gtilde_cov, 0, 1, tkij_a *
                (tkij_a+tkij_c), 1, 3) ;

    source_Sij += - 2. * qpig * nn * ( psi4 * star_i.stress_euler 
              - 0.33333333333333333 * star_i.s_euler * gtilde_con ) ; 
        
    source_Sij += - 1./(psi4*psi2) * contract(gtilde_con, 1, 
            contract(gtilde_con, 1, qq*dcovdcov_lnq_auto + 
                 qq*dcov_lnq_auto*dcov_lnq, 0), 1) ;

    source_Sij += - 0.5/(psi4*psi2) * contract(contract(star_i.hij, 1,
                    dcov_hij_auto, 2), 1, dcov_qq, 0) -
        0.5/(psi4*psi2) * contract(contract(dcov_hij_auto, 2,
                    star_i.hij, 1), 1, dcov_qq, 0) ;
                    
    source_Sij += 0.5/(psi4*psi2) * contract(contract(star_i.hij, 0,
                     dcov_hij_auto, 2), 0, dcov_qq, 0) ;

    source_Sij += 1./(3.*psi4*psi2)*contract(gtilde_con, 0, 1,
       qq*dcovdcov_lnq_auto + qq*dcov_lnq_auto*dcov_lnq, 0, 1)
        *gtilde_con ;

    source_Sij += 1./(3.*psi4*psi2) * contract(hdirac, 0, 
                        dcov_qq, 0)*star_i.hij_auto ;

    // Source terms for Rij
    //---------------------

    source_Rij = contract(star_i.hij, 0, 1, dcov_hij_auto.derive_cov(flat),
                  2, 3) ;
    source_Rij.inc_dzpuis() ;


    source_Rij += - contract(star_i.hij_auto, 1, dcov_hdirac, 1) -
        contract(dcov_hdirac, 1, star_i.hij_auto, 1) ;
        
    source_Rij += contract(hdirac, 0, dcov_hij_auto, 2) ;

    source_Rij += - contract(contract(dcov_hij_auto, 1, dcov_hij, 2),
                 1, 3) ;

    source_Rij += - contract(gtilde_cov, 0, 1, contract(contract(
        gtilde_con, 0, dcov_hij_auto, 2), 0, dcov_hij, 2), 1, 3) ;

    source_Rij += contract(contract(contract(contract(gtilde_cov, 0, 
          dcov_hij_auto, 1), 2, gtilde_con, 1), 0, dcov_hij, 1), 0, 3) +
        contract(contract(contract(contract(gtilde_cov, 0, 
        dcov_hij_auto, 1), 0, dcov_hij, 1), 0, 3), 0, gtilde_con, 1) ;

    source_Rij += 0.5 * contract(gtilde_con*gtilde_con, 1, 3, 
         contract(dcov_hij_auto, 0, 1, dcov_gtilde, 0, 1), 0, 1) ;

    source_Rij = source_Rij * 0.5 ;

    for(int i=1; i<=3; i++) 
        for(int j=1; j<=i; j++) {

        source_tot_hij = source_1(i,j) + source_1(j,i) 
            + source_2(i,j) + 2.*psi4/nn * (
            source_4(i,j) - source_Sij(i,j)) 
            - 2.* source_Rij(i,j) +
            source_5(i,j) + source_5(j,i) + source_6(i,j) ;
        source_tot_hij.dec_dzpuis() ;
            
        source3 = 2.*psi4/nn * (source_3a(i,j) + source_3a(j,i)  
                 + source_3b(i,j)) ; 

        source_tot_hij = source_tot_hij + source3 ;


        if (ll == 1){
            if(i==1 && j==1) {
            Scalar lapl (star_i.hij_auto(1,1).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_1.set(1,1) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==2 && j==1) {
            Scalar lapl (star_i.hij_auto(2,1).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_1.set(2,1) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==3 && j==1) {
            Scalar lapl (star_i.hij_auto(3,1).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_1.set(3,1) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==2 && j==2) {
            Scalar lapl (star_i.hij_auto(2,2).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_1.set(2,2) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==3 && j==2) {
            Scalar lapl (star_i.hij_auto(3,2).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_1.set(3,2) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==3 && j==3) {
            Scalar lapl (star_i.hij_auto(3,3).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_1.set(3,3) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
        }

        if (ll == 2){
            if(i==1 && j==1) {
            Scalar lapl (star_i.hij_auto(1,1).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_2.set(1,1) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==2 && j==1) {
            Scalar lapl (star_i.hij_auto(2,1).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_2.set(2,1) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==3 && j==1) {
            Scalar lapl (star_i.hij_auto(3,1).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_2.set(3,1) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==2 && j==2) {
            Scalar lapl (star_i.hij_auto(2,2).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_2.set(2,2) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==3 && j==2) {
            Scalar lapl (star_i.hij_auto(3,2).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_2.set(3,2) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
            if(i==3 && j==3) {
            Scalar lapl (star_i.hij_auto(3,3).laplacian()) ;
            lapl.dec_dzpuis() ;
            lie_aij_2.set(3,3) = nn / (2.*psi4) * 
                (lapl-source_tot_hij) ;
            }
        }
        }
    }

    lie_aij_1.dec_dzpuis(3) ;
    lie_aij_2.dec_dzpuis(3) ;
        
    int nz = star1.mp.get_mg()->get_nzone() ;
  
    // Construction of an auxiliar grid and mapping (Last domain is at lambda)
    double* bornes = new double [6] ;
    bornes[nz] = __infinity ;
    bornes[4] = M_PI / omega ;
    bornes[3] = M_PI / omega * 0.5 ;
    bornes[2] = M_PI / omega  * 0.2 ;
    bornes[1] = M_PI / omega  * 0.1 ;
    bornes[0] = 0 ;
    
    Map_af mapping (*(star1.mp.get_mg()), bornes) ;
    
    delete [] bornes ; 
 
    
    Sym_tensor lie_aij2_1 (star1.mp, CON, star1.mp.get_bvect_cart()) ;
    Sym_tensor lie_aij_tot_1 (star1.mp, CON, star1.mp.get_bvect_cart()) ;
    Sym_tensor lie_aij_tot (mapping, CON, mapping.get_bvect_cart()) ;

    Scalar lie_K2_1 (star1.mp) ;
    lie_K2_1.set_etat_qcq() ;
    Scalar lie_K_tot_1 (star1.mp) ;


    // Importation on the mapping 1
    // -------------------------------

    lie_K2_1.import(lie_K_2) ;
    lie_K_tot_1 = lie_K_1 + lie_K2_1 ;
    lie_K_tot_1.inc_dzpuis(2) ;

    lie_aij_2.change_triad(star1.mp.get_bvect_cart()) ;    
    for(int i=1; i<=3; i++) 
    for(int j=1; j<=i; j++) {
        lie_aij2_1.set(i,j).import(lie_aij_2(i,j)) ;
        lie_aij2_1.set(i,j).set_spectral_va().set_base(lie_aij_2(i,j).
                        get_spectral_va().get_base()) ;
    }

    lie_aij_tot_1 = lie_aij_1 + lie_aij2_1 ;
    lie_aij_tot_1.inc_dzpuis(2) ;

    
    Sym_tensor lie_kij_tot (star1.mp, CON, star1.mp.get_bvect_cart()) ;
    lie_kij_tot = lie_aij_tot_1/star1.psi4 + 1./3.*star1.gamma.con()*
    lie_K_tot_1 ;


    cout << " IN THE CENTER OF STAR 1 " << endl 
     << " ----------------------- " << endl ;
    /*
    cout << " components xx, xy, yy, xz, yz, zz" << endl ;
    for(int i=1; i<=3; i++) 
    for(int j=1; j<=i; j++) {
        Scalar resu(lie_kij_tot(i,j)*lie_kij_tot(i,j)) ;
        cout << "i = " << i << ", j = " << j << endl ; 
        cout << "norme de la diff " << endl 
         << norme(resu)/(nr*nt*np) << endl ;

        // Computation of the integral
        // -----------------------------

        Tbl integral (nz) ;
        integral.annule_hard()  ;
        Tbl integ (resu.integrale_domains()) ;
        for (int mm=0; mm<nz; mm++) 
        for (int pp=0; pp<=mm; pp++) 
            integral.set(mm) += integ(pp) ; 
        cout << sqrt(integral) / sqrt(omega) << endl ;  // To get 
        // dimensionless quantity
    }
    */

    cout << "L2 norm of L_k K^{ab} " << endl ;
    Scalar determinant (pow(star1.get_gamma().determinant(), 0.5)) ;
    determinant.std_spectral_base() ;
    Scalar resu(2.*contract(lie_kij_tot, 0, 1, 
             lie_kij_tot.up_down(star1.gamma), 0, 1)
        *determinant) ;
    Tbl integral (nz) ;
    integral.annule_hard() ;
    Tbl integ (resu.integrale_domains()) ;
    for (int mm=0; mm<nz; mm++) 
    for (int pp=0; pp<=mm; pp++) 
        integral.set(mm) += integ(pp) ; 
    cout << sqrt(integral) * sqrt(mass_adm()*ggrav) << endl ;  
    cout << sqrt(integral) / sqrt(omega) << endl ;  // To get 
    // dimensionless quantity

    
    cout << "omega = " << omega << endl ;
    cout << "mass_adm = " << mass_adm() << endl ;
    

    lie_kij_tot.dec_dzpuis(2) ;
    
    cout << "Position du centre de l'etoile x/lambda = "
     << -star1.get_mp().get_ori_x() * omega / M_PI << " in Lorene units"
     << endl ;
   


    // Importation on the mapping defined in the center of mass
    // -------------------------------------------------------------
/*
    lie_aij_tot_1.change_triad(mapping.get_bvect_cart()) ;    
    for(int i=1; i<=3; i++) 
    for(int j=1; j<=i; j++) {
        lie_aij_tot.set(i,j).import(lie_aij_tot_1(i,j)) ;
        lie_aij_tot.set(i,j).set_spectral_va().set_base(lie_aij_tot_1(i,j).
                        get_spectral_va().get_base()) ;
    }

    lie_aij_tot.inc_dzpuis(2) ;

    cout << " IN THE CENTER OF MASS : " << endl 
     << " ----------------------- " << endl ;

    cout << " components xx, xy, yy, xz, yz, zz" << endl ;
    for(int i=1; i<=3; i++) 
    for(int j=1; j<=i; j++) {
        Scalar resu(lie_aij_tot(i,j)*lie_aij_tot(i,j)) ;
        cout << "i = " << i << ", j = " << j << endl ; 
        cout << "norme de la diff " << endl 
         << norme(resu)/(nr*nt*np) << endl ;

        Tbl integral (nz) ;
        integral.annule_hard()  ;
        Tbl integ (resu.integrale_domains()) ;
        for (int mm=0; mm<nz; mm++) 
        for (int pp=0; pp<=mm; pp++) 
            integral.set(mm) += integ(pp) ; 
        cout << sqrt(integral) / sqrt(omega) << endl ;  // To get 
        // dimensionless quantity
    }

    cout << "L2 norm of L_k K^{ab} " << endl ;
    resu = contract(lie_aij_tot, 0, 1, 
            lie_aij_tot.up_down(star1.gtilde), 0, 1) ;
    integral.annule_hard()  ;
    integ = resu.integrale_domains() ;
    for (int mm=0; mm<nz; mm++) 
    for (int pp=0; pp<=mm; pp++) 
        integral.set(mm) += integ(pp) ; 
    cout << sqrt(integral) / sqrt(omega) << endl ;  // To get 
    // dimensionless quantity
    */

    cout << "Omega M = " << omega * mass_adm()*ggrav << endl ;

}

---