et_bin_equilibrium.C

/*
 *  Method of class Etoile_bin to compute an equilibrium configuration
 *
 *  (see file etoile.h for documentation).
 *
 */

/*
 *   Copyright (c) 2000-2001 Eric Gourgoulhon
 *   Copyright (c) 2000-2001 Keisuke Taniguchi
 *
 *   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 et_bin_equilibrium_C[] = "$Header: /cvsroot/Lorene/C++/Source/Etoile/et_bin_equilibrium.C,v 1.12 2009/06/15 09:25:18 k_taniguchi Exp $" ;

/*
 * $Id: et_bin_equilibrium.C,v 1.12 2009/06/15 09:25:18 k_taniguchi Exp $
 * $Log: et_bin_equilibrium.C,v $
 * Revision 1.12  2009/06/15 09:25:18  k_taniguchi
 * Improved the rescaling of the domains.
 *
 * Revision 1.11  2008/11/14 13:48:06  e_gourgoulhon
 * Added parameter pent_limit to force the enthalpy values at the
 * boundaries between the domains, in case of more than one domain inside
 * the star.
 *
 * Revision 1.10  2004/09/28 15:49:23  f_limousin
 * Improve the rescaling of the domains for nzone = 4 and nzone = 5.
 *
 * Revision 1.9  2004/05/13 08:47:01  f_limousin
 * Decomment the procedure resize.
 *
 * Revision 1.8  2004/05/10 10:15:57  f_limousin
 * Change to avoid a warning in the compilation of Lorene
 *
 * Revision 1.7  2004/05/07 12:36:34  f_limousin
 * Add new member ssjm1_psi in order to have only one function
 * equilibrium (the same for strange stars and neutron stars)
 *
 * Revision 1.6  2004/05/07 08:32:44  k_taniguchi
 * Introduction of the version without ssjm1_psi.
 *
 * Revision 1.5  2004/04/19 11:06:36  f_limousin
 * Differents call of Etoile_bin::velocity_potential depending on
 * the equation of state.
 *
 * Revision 1.4  2004/03/25 10:29:04  j_novak
 * All LORENE's units are now defined in the namespace Unites (in file unites.h).
 *
 * Revision 1.3  2003/01/17 13:31:13  f_limousin
 * Add comments
 *
 * Revision 1.2  2002/12/10 13:28:03  k_taniguchi
 * Change the multiplication "*" to "%"
 *   and flat_scalar_prod to flat_scalar_prod_desal.
 *
 * Revision 1.1.1.1  2001/11/20 15:19:28  e_gourgoulhon
 * LORENE
 *
 * Revision 2.23  2001/08/07  09:43:02  keisuke
 * Change of the method to set the longest radius of a star
 *  on the first domain.
 * Addition of a new argument in Etoile_bin::equilibrium : Tbl fact.
 *
 * Revision 2.22  2001/06/22  08:54:19  keisuke
 * Set the inner values of the second domain of ent
 *   by using the outer ones of the first domain.
 *
 * Revision 2.21  2001/06/18  12:50:49  keisuke
 * Addition of the filter for the source of shift vector.
 *
 * Revision 2.20  2001/05/17  12:18:47  keisuke
 * Change of the method to calculate chi from setting position in map
 *  to val_point.
 *
 * Revision 2.19  2001/01/16  17:01:53  keisuke
 * Change the method to set the values on the surface.
 *
 * Revision 2.18  2001/01/10  16:42:51  keisuke
 * Set the inner values of the second domain of logn_auto
 *   by using the outer ones of the first domain.
 *
 * Revision 2.17  2000/12/22  13:08:04  eric
 * precis_adapt = 1e-14 au lieu de 1e-15.
 * Sorties graphiques commentees.
 *
 * Revision 2.16  2000/12/20  10:32:44  eric
 * Changement important : nz_search = nzet ---> nz_search = nzet + 1
 *
 * Revision 2.15  2000/10/23  14:02:16  eric
 * Modif de Map_et::adapt: on y rentre desormais avec nz_search
 *  (dans le cas present nz_search = nzet).
 *
 * Revision 2.14  2000/09/28  12:19:36  keisuke
 * Construct logn_auto_regu from logn_auto.
 * This procedure is needed for et_bin_upmetr.C.
 *
 * Revision 2.13  2000/05/25  13:48:12  eric
 * Ajout de l'argument thres_adapt: l'adaptation du mapping n'est
 * plus effectuee si dH/dr_eq passe sous un certain seuil.
 *
 * Revision 2.12  2000/05/25  12:58:31  eric
 * Modifs classe Param: les int et double sont desormais passes par leurs
 *  adresses.
 *
 * Revision 2.11  2000/03/29  11:57:38  eric
 * *** empty log message ***
 *
 * Revision 2.10  2000/03/29  11:53:41  eric
 * Modif affichage
 *
 * Revision 2.9  2000/03/22  16:37:29  eric
 * Calcul des erreurs dans la resolution des equations de Poisson
 * et sortie de ces erreurs dans le Tbl diff.
 *
 * Revision 2.8  2000/03/22  12:56:18  eric
 * Nouveau prototype d'Etoile_bin::equilibrium : diff_ent est remplace
 *   par le Tbl diff.
 *
 * Revision 2.7  2000/03/10  15:47:19  eric
 * On appel desormais poisson_vect avec dzpuis = 4.
 *
 * Revision 2.6  2000/03/07  16:52:15  eric
 * Modifs manipulations source pour le shift.
 *
 * Revision 2.5  2000/03/07  08:32:47  eric
 * Appel de Map_radial::reevaluate_sym (pour tenir compte de la symetrie
 *  / plan y=0).
 *
 * Revision 2.4  2000/02/17  19:56:57  eric
 * L'appel de Map_radial::reevaluate pour ent est fait sur nzet+1 zone
 * et non plus nzet.
 *
 * Revision 2.2  2000/02/16  17:12:03  eric
 * Premiere version avec les equations du champ gravitationnel.
 *
 * Revision 2.1  2000/02/15  15:59:52  eric
 * *** empty log message ***
 *
 * Revision 2.0  2000/02/15  15:40:42  eric
 * *** empty log message ***
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Etoile/et_bin_equilibrium.C,v 1.12 2009/06/15 09:25:18 k_taniguchi Exp $
 *
 */

// Headers C
#include <math.h>

// Headers Lorene
#include "etoile.h"
#include "param.h"
#include "eos.h" 

#include "graphique.h"
#include "utilitaires.h"
#include "unites.h"     

void Etoile_bin::equilibrium(double ent_c, 
                             int mermax, int mermax_poisson, 
             double relax_poisson, int mermax_potvit, 
             double relax_potvit, double thres_adapt,
             const Tbl& fact_resize, Tbl& diff, const Tbl* pent_limit ) {
                 

    // Fundamental constants and units
    // -------------------------------

  using namespace Unites ;
    
    // Initializations
    // ---------------
    
    const Mg3d* mg = mp.get_mg() ; 
    int nz = mg->get_nzone() ;      // total number of domains
    
    // The following is required to initialize mp_prev as a Map_et:
    Map_et& mp_et = dynamic_cast<Map_et&>(mp) ; 
    
    // Domain and radial indices of points at the surface of the star:
    int l_b = nzet - 1 ; 
    int i_b = mg->get_nr(l_b) - 1 ; 
    int k_b ;
    int j_b ; 
    
    // Value of the enthalpy defining the surface of the star
    double ent_b = 0 ; 
    
    // Error indicators
    // ----------------
    
    double& diff_ent = diff.set(0) ; 
    double& diff_vel_pot = diff.set(1) ; 
    double& diff_logn = diff.set(2) ; 
    double& diff_beta = diff.set(3) ; 
    double& diff_shift_x = diff.set(4) ; 
    double& diff_shift_y = diff.set(5) ; 
    double& diff_shift_z = diff.set(6) ; 

    // Parameters for the function Map_et::adapt
    // -----------------------------------------
    
    Param par_adapt ; 
    int nitermax = 100 ;  
    int niter ; 
    int adapt_flag = 1 ;    //  1 = performs the full computation, 
                //  0 = performs only the rescaling by 
                //      the factor alpha_r
    //##    int nz_search = nzet + 1 ;  // Number of domains for searching the enthalpy
    int nz_search = nzet ;  // Number of domains for searching the enthalpy
                //  isosurfaces

    double precis_secant = 1.e-14 ; 
    double alpha_r ; 
    double reg_map = 1. ; // 1 = regular mapping, 0 = contracting mapping

    par_adapt.add_int(nitermax, 0) ; // maximum number of iterations to 
                     // locate zeros by the secant method
    par_adapt.add_int(nzet, 1) ;    // number of domains where the adjustment 
                    // to the isosurfaces of ent is to be 
                    // performed
    par_adapt.add_int(nz_search, 2) ;   // number of domains to search for
                    // the enthalpy isosurface
    par_adapt.add_int(adapt_flag, 3) ; //  1 = performs the full computation, 
                       //  0 = performs only the rescaling by 
                       //      the factor alpha_r
    par_adapt.add_int(j_b, 4) ; //  theta index of the collocation point 
                    //  (theta_*, phi_*)
    par_adapt.add_int(k_b, 5) ; //  theta index of the collocation point 
                    //  (theta_*, phi_*)

    par_adapt.add_int_mod(niter, 0) ;  //  number of iterations actually used in 
                       //  the secant method
    
    par_adapt.add_double(precis_secant, 0) ; // required absolute precision in 
                         // the determination of zeros by 
                         // the secant method
    par_adapt.add_double(reg_map, 1)    ;  // 1. = regular mapping, 0 = contracting mapping
    
    par_adapt.add_double(alpha_r, 2) ;      // factor by which all the radial 
                        // distances will be multiplied 

    // Enthalpy values for the adaptation
    Tbl ent_limit(nzet) ; 
    if (pent_limit != 0x0) ent_limit = *pent_limit ; 
           
    par_adapt.add_tbl(ent_limit, 0) ;   // array of values of the field ent 
                        // to define the isosurfaces.              
 
    // Parameters for the function Map_et::poisson for logn_auto
    // ---------------------------------------------------------

    double precis_poisson = 1.e-16 ;     

    Param par_poisson1 ; 

    par_poisson1.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_poisson1.add_double(relax_poisson,  0) ; // relaxation parameter
    par_poisson1.add_double(precis_poisson, 1) ; // required precision
    par_poisson1.add_int_mod(niter, 0) ;  //  number of iterations actually used 
    par_poisson1.add_cmp_mod( ssjm1_logn ) ; 
                       
    // Parameters for the function Map_et::poisson for beta_auto
    // ---------------------------------------------------------

    Param par_poisson2 ; 

    par_poisson2.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_poisson2.add_double(relax_poisson,  0) ; // relaxation parameter
    par_poisson2.add_double(precis_poisson, 1) ; // required precision
    par_poisson2.add_int_mod(niter, 0) ;  //  number of iterations actually used 
    par_poisson2.add_cmp_mod( ssjm1_beta ) ; 
    
                       
    // Parameters for the function Tenseur::poisson_vect
    // -------------------------------------------------

    Param par_poisson_vect ; 

    par_poisson_vect.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_poisson_vect.add_double(relax_poisson,  0) ; // relaxation parameter
    par_poisson_vect.add_double(precis_poisson, 1) ; // required precision
    par_poisson_vect.add_cmp_mod( ssjm1_khi ) ; 
    par_poisson_vect.add_tenseur_mod( ssjm1_wshift ) ; 
    par_poisson_vect.add_int_mod(niter, 0) ;   

                       
    // External potential
    // See Eq (99) from Gourgoulhon et al. (2001)
    // -----------------------------------------
    
    Tenseur pot_ext = logn_comp + pot_centri + loggam ;
//##
//  des_coupe_z(pot_ext(), 0., 1, "pot_ext", &(ent()) ) ; 
//##
    
    Tenseur ent_jm1 = ent ; // Enthalpy at previous step
    
    Tenseur source(mp) ;    // source term in the equation for logn_auto
                // and beta_auto
                
    Tenseur source_shift(mp, 1, CON, ref_triad) ;  // source term in the equation 
                           //  for shift_auto

    //=========================================================================
    //          Start of iteration
    //=========================================================================

    for(int mer=0 ; mer<mermax ; mer++ ) {

    cout << "-----------------------------------------------" << endl ;
    cout << "step: " << mer << endl ;
    cout << "diff_ent = " << diff_ent << endl ;    

    //-----------------------------------------------------
    // Resolution of the elliptic equation for the velocity
    // scalar potential
    //-----------------------------------------------------

    if (irrotational) {
        diff_vel_pot = velocity_potential(mermax_potvit, 
                           precis_poisson, relax_potvit) ; 

    }

    //-----------------------------------------------------
    // Computation of the new radial scale
    //-----------------------------------------------------

    // alpha_r (r = alpha_r r') is determined so that the enthalpy
    // takes the requested value ent_b at the stellar surface
    
    // Values at the center of the star:
    double logn_auto_c  = logn_auto()(0, 0, 0, 0) ; 
    double pot_ext_c  = pot_ext()(0, 0, 0, 0) ; 

    // Search for the reference point (theta_*, phi_*) [notation of
    //  Bonazzola, Gourgoulhon & Marck PRD 58, 104020 (1998)]
    //  at the surface of the star
    // ------------------------------------------------------------
    double alpha_r2 = 0 ; 
    for (int k=0; k<mg->get_np(l_b); k++) {
        for (int j=0; j<mg->get_nt(l_b); j++) {
        
        double pot_ext_b  = pot_ext()(l_b, k, j, i_b) ; 
        double logn_auto_b  = logn_auto()(l_b, k, j, i_b) ; 


        // See Eq (100) from Gourgoulhon et al. (2001)
        double alpha_r2_jk = ( ent_c - ent_b + pot_ext_c - pot_ext_b) / 
                ( logn_auto_b - logn_auto_c ) ;
        
//      cout << "k, j, alpha_r2_jk : " << k << "  " << j << "  " 
//           << alpha_r2_jk << endl ; 
          
        if (alpha_r2_jk > alpha_r2) {
            alpha_r2 = alpha_r2_jk ; 
            k_b = k ; 
            j_b = j ; 
        }

        }
    }
    
    alpha_r = sqrt(alpha_r2) ;
        
    cout << "k_b, j_b, alpha_r: " << k_b << "  " << j_b << "  " 
         <<  alpha_r << endl ;

    // New value of logn_auto 
    // ----------------------
    
    logn_auto = alpha_r2 * logn_auto ;
    logn_auto_regu = alpha_r2 * logn_auto_regu ;
    logn_auto_c  = logn_auto()(0, 0, 0, 0) ;


    //------------------------------------------------------------
    // Change the values of the inner points of the second domain
    // by those of the outer points of the first domain
    //------------------------------------------------------------

    (logn_auto().va).smooth(nzet, (logn_auto.set()).va) ;


    //------------------------------------------
    // First integral   --> enthalpy in all space
    // See Eq (98) from Gourgoulhon et al. (2001)
    //-------------------------------------------

    ent = (ent_c + logn_auto_c + pot_ext_c) - logn_auto - pot_ext ;

    (ent().va).smooth(nzet, (ent.set()).va) ;

    //----------------------------------------------------
    // Adaptation of the mapping to the new enthalpy field
    //----------------------------------------------------
    
    // Shall the adaptation be performed (cusp) ?
    // ------------------------------------------
    
    double dent_eq = ent().dsdr().val_point(ray_eq(),M_PI/2.,0.) ;
    double dent_pole = ent().dsdr().val_point(ray_pole(),0.,0.) ;
    double rap_dent = fabs( dent_eq / dent_pole ) ; 
    cout << "| dH/dr_eq / dH/dr_pole | = " << rap_dent << endl ; 
    
    if ( rap_dent < thres_adapt ) {
        adapt_flag = 0 ;    // No adaptation of the mapping 
        cout << "******* FROZEN MAPPING  *********" << endl ; 
    }
    else{
        adapt_flag = 1 ;    // The adaptation of the mapping is to be
                //  performed
    }


    if (pent_limit == 0x0) {
      ent_limit.set_etat_qcq() ; 
      for (int l=0; l<nzet; l++) {  // loop on domains inside the star
        ent_limit.set(l) = ent()(l, k_b, j_b, i_b) ; 
      }
      ent_limit.set(nzet-1) = ent_b  ; 
    }

    Map_et mp_prev = mp_et ; 

//##    cout  << "Enthalpy field at the outer boundary of domain 0 : " 
//   << endl ; 
//    for (int k=0; k<mg->get_np(0); k++) {
//  cout << "k = " << k << " : " ; 
//  for (int j=0; j<mg->get_nt(0); j++) {
//      cout << "  " << ent()(0, k, j, mg->get_nr(0)-1) ;
//  }
//  cout << endl ; 
//  }
//    cout  << "Enthalpy field at the inner boundary of domain 1 : " 
//   << endl ; 
//    for (int k=0; k<mg->get_np(1); k++) {
//  cout << "k = " << k << " : " ; 
//  for (int j=0; j<mg->get_nt(1); j++) {
//      cout << "  " << ent()(1, k, j, 0) ;
//  }
//  cout << endl ; 
//    }
//    cout  << "Difference enthalpy field boundary between domains 0 and 1: " 
//   << endl ; 
//    for (int k=0; k<mg->get_np(1); k++) {
//  cout << "k = " << k << " : " ; 
//  for (int j=0; j<mg->get_nt(1); j++) {
//      cout << "  " << ent()(0, k, j, mg->get_nr(0)-1) -
//              ent()(1, k, j, 0) ;
//  }
//  cout << endl ; 
//    }


//##
//  des_coupe_z(gam_euler(), 0., 1, "gam_euler") ; 
//  des_coupe_z(loggam(), 0., 1, "loggam") ; 
//  des_coupe_y(loggam(), 0., 1, "loggam") ; 
//  des_coupe_z(d_psi(0), 0., 1, "d_psi_0") ; 
//  des_coupe_z(d_psi(1), 0., 1, "d_psi_1") ; 
//  des_coupe_z(d_psi(2), 0., 1, "d_psi_2") ; 
//  des_coupe_z(ent(), 0., 1, "ent before adapt", &(ent()) ) ; 
//##


    mp.adapt(ent(), par_adapt) ;

    // Readjustment of the external boundary of domain l=nzet
    // to keep a fixed ratio with respect to star's surface
    
    double rr_in_1 = mp.val_r(nzet, -1., M_PI/2., 0.) ;

    // Resizes the outer boundary of the shell including the comp. NS
    double rr_out_nm2 = mp.val_r(nz-2, 1., M_PI/2., 0.) ;

    mp.resize(nz-2, rr_in_1/rr_out_nm2 * fact_resize(1)) ;

    // Resizes the inner boundary of the shell including the comp. NS
    double rr_out_nm3 = mp.val_r(nz-3, 1., M_PI/2., 0.) ;

    mp.resize(nz-3, rr_in_1/rr_out_nm3 * fact_resize(0)) ;

    if (nz > nzet+3) {

        // Resize of the domain from 1(nzet) to N-4
        double rr_out_nm3_new = mp.val_r(nz-3, 1., M_PI/2., 0.) ;

        for (int i=nzet-1; i<nz-4; i++) {

            double rr_out_i = mp.val_r(i, 1., M_PI/2., 0.) ;

        double rr_mid = rr_out_i
          + (rr_out_nm3_new - rr_out_i) / double(nz - 3 - i) ;

        double rr_2timesi = 2. * rr_out_i ;

        if (rr_2timesi < rr_mid) {

            double rr_out_ip1 = mp.val_r(i+1, 1., M_PI/2., 0.) ;

            mp.resize(i+1, rr_2timesi / rr_out_ip1) ;

        }
        else {

            double rr_out_ip1 = mp.val_r(i+1, 1., M_PI/2., 0.) ;

            mp.resize(i+1, rr_mid / rr_out_ip1) ;

        }  // End of else

        }  // End of i loop

    }  // End of (nz > nzet+3) loop


    //----------------------------------------------------
    // Computation of the enthalpy at the new grid points
    //----------------------------------------------------

    mp_prev.homothetie(alpha_r) ; 
    
    mp.reevaluate_symy(&mp_prev, nzet+1, ent.set()) ; 

//  des_coupe_z(ent(), 0., 1, "ent after reevaluate", &(ent()) ) ; 

    double ent_s_max = -1 ; 
    int k_s_max = -1 ; 
    int j_s_max = -1 ; 
    for (int k=0; k<mg->get_np(l_b); k++) {
        for (int j=0; j<mg->get_nt(l_b); j++) {
        double xx = fabs( ent()(l_b, k, j, i_b) ) ;
        if (xx > ent_s_max) {
            ent_s_max = xx ; 
            k_s_max = k ; 
            j_s_max = j ; 
        }
        }
    }
    cout << "Max. abs(enthalpy) at the boundary between domains nzet-1"
         << " and nzet : " << endl ; 
    cout << "   " << ent_s_max << " reached for k = " << k_s_max <<
        " and j = " << j_s_max << endl ; 

    //----------------------------------------------------
    // Equation of state  
    //----------------------------------------------------
    
    equation_of_state() ;   // computes new values for nbar (n), ener (e) 
                // and press (p) from the new ent (H)
    
    //---------------------------------------------------------
    // Matter source terms in the gravitational field equations 
    //---------------------------------------------------------

    hydro_euler() ;     // computes new values for ener_euler (E), 
                // s_euler (S) and u_euler (U^i)

    //--------------------------------------------------------
    // Poisson equation for logn_auto (nu_auto)
    //--------------------------------------------------------

    // Source 
    // See Eq (50) from Gourgoulhon et al. (2001)
    // ------------------------------------------
    
    if (relativistic) {
        source = qpig * a_car % (ener_euler + s_euler)
            + akcar_auto + akcar_comp 
         - flat_scalar_prod_desal(d_logn_auto,
                 d_beta_auto + d_beta_comp) ;
     
    }
    else {
        source = qpig * nbar ; 
    }
    
    source.set_std_base() ;     

    // Resolution of the Poisson equation 
    // ----------------------------------

    source().poisson(par_poisson1, logn_auto.set()) ; 

    // Construct logn_auto_regu for et_bin_upmetr.C
    // --------------------------------------------

    logn_auto_regu = logn_auto ;

    // Check: has the Poisson equation been correctly solved ?
    // -----------------------------------------------------
    
    Tbl tdiff_logn = diffrel(logn_auto().laplacien(), source()) ;
    cout << 
    "Relative error in the resolution of the equation for logn_auto : "
    << endl ; 
    for (int l=0; l<nz; l++) {
        cout << tdiff_logn(l) << "  " ; 
    }
    cout << endl ;
    diff_logn = max(abs(tdiff_logn)) ; 


    if (relativistic) {

        //--------------------------------------------------------
        // Poisson equation for beta_auto 
        //--------------------------------------------------------

        // Source 
        // See Eq (51) from Gourgoulhon et al. (2001)
        // ------------------------------------------
    
        source = qpig * a_car % s_euler
            + .75 * ( akcar_auto + akcar_comp )
            - .5 * flat_scalar_prod_desal(d_logn_auto,  
                          d_logn_auto + d_logn_comp)
            - .5 * flat_scalar_prod_desal(d_beta_auto,  
                          d_beta_auto + d_beta_comp) ;
             
        source.set_std_base() ;     

        // Resolution of the Poisson equation 
        // ----------------------------------

        source().poisson(par_poisson2, beta_auto.set()) ; 

        // Check: has the Poisson equation been correctly solved ?
        // -----------------------------------------------------
    
        Tbl tdiff_beta = diffrel(beta_auto().laplacien(), source()) ;
        cout << 
        "Relative error in the resolution of the equation for beta_auto : "
        << endl ; 
        for (int l=0; l<nz; l++) {
        cout << tdiff_beta(l) << "  " ; 
        }
        cout << endl ;
        diff_beta = max(abs(tdiff_beta)) ; 

        //--------------------------------------------------------
        // Vector Poisson equation for shift_auto 
        //--------------------------------------------------------

        // Source
        // See Eq (52) from Gourgoulhon et al. (2001)
        // ------
        
        Tenseur vtmp =  6. * ( d_beta_auto + d_beta_comp )
               -8. * ( d_logn_auto + d_logn_comp ) ;
        
        source_shift = (-4.*qpig) * nnn % a_car % (ener_euler + press)
          % u_euler 
        + nnn % flat_scalar_prod_desal(tkij_auto, vtmp) ;
    
        source_shift.set_std_base() ;   
 
        // Resolution of the Poisson equation 
        // ----------------------------------

        // Filter for the source of shift vector
        
        for (int i=0; i<3; i++) {

          if (source_shift(i).get_etat() != ETATZERO)
            source_shift.set(i).filtre(4) ;

        }
        
        // For Tenseur::poisson_vect, the triad must be the mapping triad,
        // not the reference one:
        
        source_shift.change_triad( mp.get_bvect_cart() ) ; 

        for (int i=0; i<3; i++) {
        if(source_shift(i).dz_nonzero()) {
            assert( source_shift(i).get_dzpuis() == 4 ) ; 
        }
        else{
            (source_shift.set(i)).set_dzpuis(4) ; 
        }
        }

        //##
        // source_shift.dec2_dzpuis() ;    // dzpuis 4 -> 2

        double lambda_shift = double(1) / double(3) ; 
    
        source_shift.poisson_vect(lambda_shift, par_poisson_vect, 
                          shift_auto, w_shift, khi_shift) ;      


        // Check: has the equation for shift_auto been correctly solved ?
        // --------------------------------------------------------------
        
        // Divergence of shift_auto : 
        Tenseur divn = contract(shift_auto.gradient(), 0, 1) ; 
        divn.dec2_dzpuis() ;    // dzpuis 2 -> 0
        
        // Grad(div) : 
        Tenseur graddivn = divn.gradient() ; 
        graddivn.inc2_dzpuis() ;    // dzpuis 2 -> 4
        
        // Full operator : 
        Tenseur lap_shift(mp, 1, CON, mp.get_bvect_cart() ) ;  
        lap_shift.set_etat_qcq() ; 
        for (int i=0; i<3; i++) {
        lap_shift.set(i) = shift_auto(i).laplacien() 
                    + lambda_shift * graddivn(i) ;
        }

        Tbl tdiff_shift_x = diffrel(lap_shift(0), source_shift(0)) ; 
        Tbl tdiff_shift_y = diffrel(lap_shift(1), source_shift(1)) ; 
        Tbl tdiff_shift_z = diffrel(lap_shift(2), source_shift(2)) ; 

        cout << 
        "Relative error in the resolution of the equation for shift_auto : "
        << endl ; 
        cout << "x component : " ;
        for (int l=0; l<nz; l++) {
        cout << tdiff_shift_x(l) << "  " ; 
        }
        cout << endl ;
        cout << "y component : " ;
        for (int l=0; l<nz; l++) {
        cout << tdiff_shift_y(l) << "  " ; 
        }
        cout << endl ;
        cout << "z component : " ;
        for (int l=0; l<nz; l++) {
        cout << tdiff_shift_z(l) << "  " ; 
        }
        cout << endl ;
            
        diff_shift_x = max(abs(tdiff_shift_x)) ; 
        diff_shift_y = max(abs(tdiff_shift_y)) ; 
        diff_shift_z = max(abs(tdiff_shift_z)) ; 

        // Final result
        // ------------
        // The output of Tenseur::poisson_vect is on the mapping triad,
        // it should therefore be transformed to components on the
        // reference triad :
        
        shift_auto.change_triad( ref_triad ) ; 


    }   // End of relativistic equations    
    

    if (nzet > 1) {
      cout.precision(10) ;

      for (int ltrans = 0; ltrans < nzet-1; ltrans++) {
    cout << endl << "Values at boundary between domains no. " << ltrans << " and " <<   ltrans+1 << " for theta = pi/2 and phi = 0 :" << endl ;

    double rt1 = mp.val_r(ltrans, 1., M_PI/2, 0.) ; 
    double rt2 = mp.val_r(ltrans+1, -1., M_PI/2, 0.) ; 
    cout << "   Coord. r [km] (left, right, rel. diff) : " 
      << rt1 / km << "  " << rt2 / km << "  " << (rt2 - rt1)/rt1  << endl ; 
  
    int ntm1 = mg->get_nt(ltrans) - 1; 
    int nrm1 = mg->get_nr(ltrans) - 1 ; 
    double ent1 = ent()(ltrans, 0, ntm1, nrm1) ; 
    double ent2 = ent()(ltrans+1, 0, ntm1, 0) ; 
      cout << "   Enthalpy (left, right, rel. diff) : " 
        << ent1 << "  " << ent2 << "  " << (ent2-ent1)/ent1  << endl ; 

    double press1 = press()(ltrans, 0, ntm1, nrm1) ; 
    double press2 = press()(ltrans+1, 0, ntm1, 0) ; 
      cout << "   Pressure (left, right, rel. diff) : " 
      << press1 << "  " << press2 << "  " << (press2-press1)/press1 << endl ; 

    double nb1 = nbar()(ltrans, 0, ntm1, nrm1) ; 
    double nb2 = nbar()(ltrans+1, 0, ntm1, 0) ; 
      cout << "   Baryon density (left, right, rel. diff) : " 
      << nb1 << "  " << nb2 << "  " << (nb2-nb1)/nb1  << endl ; 
      }
    }
/*    if (mer % 10 == 0) {
    cout << "mer = " << mer << endl ; 
    double r_max = 1.2 * ray_eq() ; 
    des_profile(nbar(), 0., r_max, M_PI/2, 0., "n", "Baryon density") ; 
    des_profile(ener(), 0., r_max, M_PI/2, 0., "e", "Energy density") ; 
    des_profile(press(), 0., r_max, M_PI/2, 0., "p", "Pressure") ; 
    des_profile(ent(), 0., r_max, M_PI/2, 0., "H", "Enthalpy") ;   
    }*/

    //-------------------------------------------------
    //  Relative change in enthalpy
    //-------------------------------------------------

    Tbl diff_ent_tbl = diffrel( ent(), ent_jm1() ) ; 
    diff_ent = diff_ent_tbl(0) ; 
    for (int l=1; l<nzet; l++) {
        diff_ent += diff_ent_tbl(l) ; 
    }
    diff_ent /= nzet ; 
    
    
    ent_jm1 = ent ; 


    } // End of main loop
    
    //=========================================================================
    //          End of iteration
    //=========================================================================


}

---