star_bin_equilibrium.C

/*
 *   Method of class Star_bin to compute an equilibrium configuration
 *
 *  (see file star.h for documentation).
 */
/*
 *   Copyright (c) 2004 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 version 2
 *   as published by the Free Software Foundation.
 *
 *   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
 *
 */

 

/*
 * $Id: star_bin_equilibrium.C,v 1.30 2016/12/05 16:18:14 j_novak Exp $
 * $Log: star_bin_equilibrium.C,v $
 * Revision 1.30  2016/12/05 16:18:14  j_novak
 * Suppression of some global variables (file names, loch, ...) to prevent redefinitions
 *
 * Revision 1.29  2014/10/13 08:53:38  j_novak
 * Lorene classes and functions now belong to the namespace Lorene.
 *
 * Revision 1.28  2014/10/06 15:13:16  j_novak
 * Modified #include directives to use c++ syntax.
 *
 * Revision 1.27  2006/08/01 14:26:01  f_limousin
 * Display
 *
 * Revision 1.26  2006/05/31 09:26:04  f_limousin
 * Modif. of the size of the different domains
 *
 * Revision 1.25  2006/04/11 14:24:44  f_limousin
 * New version of the code : improvement of the computation of some
 * critical sources, estimation of the dirac gauge, helical symmetry...
 *
 * Revision 1.24  2005/11/03 13:27:09  f_limousin
 * Final version for the letter.
 *
 * Revision 1.23  2005/09/14 12:48:02  f_limousin
 * Comment graphical outputs.
 *
 * Revision 1.22  2005/09/14 12:30:52  f_limousin
 * Saving of fields lnq and logn in class Star.
 *
 * Revision 1.21  2005/09/13 19:38:31  f_limousin
 * Reintroduction of the resolution of the equations in cartesian coordinates.
 *
 * Revision 1.20  2005/04/08 12:36:44  f_limousin
 * Just to avoid warnings...
 *
 * Revision 1.19  2005/02/24 16:27:21  f_limousin
 * Add mermax_poisson and relax_poisson in the parameters of the function.
 *
 * Revision 1.18  2005/02/24 16:04:13  f_limousin
 * Change the name of some variables (for instance dcov_logn --> dlogn).
 * Improve the resolution of the tensorial poisson equation for hh.
 *
 * Revision 1.17  2005/02/18 13:14:18  j_novak
 * Changing of malloc/free to new/delete + suppression of some unused variables
 * (trying to avoid compilation warnings).
 *
 * Revision 1.16  2005/02/17 17:32:53  f_limousin
 * Change the name of some quantities to be consistent with other classes
 * (for instance nnn is changed to nn, shift to beta, beta to lnq...)
 *
 * Revision 1.15  2005/02/11 18:13:47  f_limousin
 * Important modification : all the poisson equations for the metric
 * quantities are now solved on an affine mapping.
 *
 * Revision 1.14  2004/12/17 16:23:19  f_limousin
 * Modif. comments.
 *
 * Revision 1.13  2004/06/22 12:49:12  f_limousin
 * Change qq, qq_auto and qq_comp to beta, beta_auto and beta_comp.
 *
 * Revision 1.12  2004/05/27 12:41:00  p_grandclement
 * correction of some shadowed variables
 *
 * Revision 1.11  2004/05/25 14:18:00  f_limousin
 * Include filters
 *
 * Revision 1.10  2004/05/10 10:26:22  f_limousin
 * Minor changes to avoid warnings in the compilation of Lorene
 *
 * Revision 1.9  2004/04/08 16:32:48  f_limousin
 * The new variable is ln(Q) instead of Q=psi^2*N. It improves the
 * convergence of the code.
 *
 * Revision 1.8  2004/03/25 10:29:26  j_novak
 * All LORENE's units are now defined in the namespace Unites (in file unites.h).
 *
 * Revision 1.7  2004/03/23 09:56:09  f_limousin
 * Many minor changes
 *
 * Revision 1.6  2004/02/27 21:16:32  e_gourgoulhon
 * Function contract_desal replaced by function contract with
 * argument desaliasing set to true.
 *
 * Revision 1.5  2004/02/27 09:51:51  f_limousin
 * Many minor changes.
 *
 * Revision 1.4  2004/02/21 17:05:13  e_gourgoulhon
 * Method Scalar::point renamed Scalar::val_grid_point.
 * Method Scalar::set_point renamed Scalar::set_grid_point.
 *
 * Revision 1.3  2004/01/20 15:17:48  f_limousin
 * First version
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Star/star_bin_equilibrium.C,v 1.30 2016/12/05 16:18:14 j_novak Exp $
 *
 */

// C headers
#include <cmath>

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


namespace Lorene {
void Star_bin::equilibrium(double ent_c, int mermax, int mermax_potvit, 
               int mermax_poisson, double relax_poisson, 
               double relax_potvit, double thres_adapt,
               Tbl& diff, double om) {

    // 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_lnq = diff.set(3) ; 
    double& diff_beta_x = diff.set(4) ; 
    double& diff_beta_y = diff.set(5) ; 
    double& diff_beta_z = diff.set(6) ; 
    double& diff_h11 = diff.set(7) ; 
    double& diff_h21 = diff.set(8) ; 
    double& diff_h31 = diff.set(9) ; 
    double& diff_h22 = diff.set(10) ; 
    double& diff_h32 = diff.set(11) ; 
    double& diff_h33 = diff.set(12) ; 



    // Parameters for te 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

    Tbl ent_limit(nz) ; 


    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  
                                        // 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 
           
    par_adapt.add_tbl(ent_limit, 0) ;   // array of values of the field ent 
                        // to define the isosurfaces. 
 

    Cmp ssjm1logn (ssjm1_logn) ;
    Cmp ssjm1lnq (ssjm1_lnq) ;
    Cmp ssjm1h11 (ssjm1_h11) ;
    Cmp ssjm1h21 (ssjm1_h21) ;
    Cmp ssjm1h31 (ssjm1_h31) ;
    Cmp ssjm1h22 (ssjm1_h22) ;
    Cmp ssjm1h32 (ssjm1_h32) ;
    Cmp ssjm1h33 (ssjm1_h33) ;
 

    ssjm1logn.set_etat_qcq() ;
    ssjm1lnq.set_etat_qcq() ;
    ssjm1h11.set_etat_qcq() ;
    ssjm1h21.set_etat_qcq() ;
    ssjm1h31.set_etat_qcq() ;
    ssjm1h22.set_etat_qcq() ;
    ssjm1h32.set_etat_qcq() ;
    ssjm1h33.set_etat_qcq() ;
   
    
    double precis_poisson = 1.e-16 ;     

    // Parameters for the function Scalar::poisson for logn_auto
    // ---------------------------------------------------------------
 
    Param par_logn ;    

    par_logn.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_logn.add_double(relax_poisson,  0) ; // relaxation parameter
    par_logn.add_double(precis_poisson, 1) ; // required precision
    par_logn.add_int_mod(niter, 0) ; // number of iterations actually used 
    par_logn.add_cmp_mod( ssjm1logn ) ; 
    
    // Parameters for the function Scalar::poisson for lnq_auto
    // ---------------------------------------------------------------
    
    Param par_lnq ; 
    
    par_lnq.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_lnq.add_double(relax_poisson,  0) ; // relaxation parameter
    par_lnq.add_double(precis_poisson, 1) ; // required precision
    par_lnq.add_int_mod(niter, 0) ; // number of iterations actually used -
    par_lnq.add_cmp_mod( ssjm1lnq ) ; 
 
    // Parameters for the function Vector::poisson for beta method 2 
    // ---------------------------------------------------------------
    
    Param par_beta2 ; 
    
    par_beta2.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_beta2.add_double(relax_poisson,  0) ; // relaxation parameter
    par_beta2.add_double(precis_poisson, 1) ; // required precision
    par_beta2.add_int_mod(niter, 0) ; // number of iterations actually used 
  
    Cmp ssjm1khi (ssjm1_khi) ;
    Tenseur ssjm1wbeta(mp, 1, CON, mp.get_bvect_cart()) ;
    ssjm1wbeta.set_etat_qcq() ;
    for (int i=0; i<3; i++) {
      ssjm1wbeta.set(i) = Cmp(ssjm1_wbeta(i+1)) ;
    }
    
    par_beta2.add_cmp_mod(ssjm1khi) ; 
    par_beta2.add_tenseur_mod(ssjm1wbeta) ; 
    
    // Parameters for the function Scalar::poisson for h11_auto
    // -------------------------------------------------------------
    
    Param par_h11 ; 
    
    par_h11.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_h11.add_double(relax_poisson,  0) ; // relaxation parameter
    par_h11.add_double(precis_poisson, 1) ; // required precision
    par_h11.add_int_mod(niter, 0) ; // number of iterations actually used 
    par_h11.add_cmp_mod( ssjm1h11 ) ; 
    
    // Parameters for the function Scalar::poisson for h21_auto
    // -------------------------------------------------------------
    
    Param par_h21 ; 
    
    par_h21.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_h21.add_double(relax_poisson,  0) ; // relaxation parameter
    par_h21.add_double(precis_poisson, 1) ; // required precision
    par_h21.add_int_mod(niter, 0) ; // number of iterations actually used 
    par_h21.add_cmp_mod( ssjm1h21 ) ; 
    
    // Parameters for the function Scalar::poisson for h31_auto
    // -------------------------------------------------------------
    
    Param par_h31 ; 
    
    par_h31.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_h31.add_double(relax_poisson,  0) ; // relaxation parameter
    par_h31.add_double(precis_poisson, 1) ; // required precision
    par_h31.add_int_mod(niter, 0) ; // number of iterations actually used 
    par_h31.add_cmp_mod( ssjm1h31 ) ; 
    
    // Parameters for the function Scalar::poisson for h22_auto
    // -------------------------------------------------------------
    
    Param par_h22 ; 
    
    par_h22.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_h22.add_double(relax_poisson,  0) ; // relaxation parameter
    par_h22.add_double(precis_poisson, 1) ; // required precision
    par_h22.add_int_mod(niter, 0) ; // number of iterations actually used 
    par_h22.add_cmp_mod( ssjm1h22 ) ; 
    
    // Parameters for the function Scalar::poisson for h32_auto
    // -------------------------------------------------------------
    
    Param par_h32 ; 
    
    par_h32.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_h32.add_double(relax_poisson,  0) ; // relaxation parameter
    par_h32.add_double(precis_poisson, 1) ; // required precision
    par_h32.add_int_mod(niter, 0) ; // number of iterations actually used 
    par_h32.add_cmp_mod( ssjm1h32 ) ; 
    
    // Parameters for the function Scalar::poisson for h33_auto
    // -------------------------------------------------------------
    
    Param par_h33 ; 
    
    par_h33.add_int(mermax_poisson,  0) ;  // maximum number of iterations
    par_h33.add_double(relax_poisson,  0) ; // relaxation parameter
    par_h33.add_double(precis_poisson, 1) ; // required precision
    par_h33.add_int_mod(niter, 0) ; // number of iterations actually used 
    par_h33.add_cmp_mod( ssjm1h33 ) ; 
    

    // External potential
    // See Eq (99) from Gourgoulhon et al. (2001)
    // ------------------
    
    cout << "logn_comp" << norme(logn_comp) << endl ;
    cout << "pot_centri" << norme(pot_centri) << endl ;
    cout << "loggam" << norme(loggam) << endl ;
    Scalar pot_ext = logn_comp + pot_centri + loggam ;
    
    Scalar ent_jm1 = ent ;  // Enthalpy at previous step
    
    Scalar source_tot(mp) ; // source term in the equation for hij_auto, 
                            // logn_auto and beta_auto
                
    Vector source_beta(mp, CON, mp.get_bvect_cart()) ;  // source term 
    // in the equation for beta_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) ; 
        
    }

    diff_vel_pot = 0. ; // to avoid the warning 

    //-----------------------------------------------------
    // 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.val_grid_point(0, 0, 0, 0) ; 
    double pot_ext_c  = pot_ext.val_grid_point(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.val_grid_point(l_b, k, j, i_b) ; 
        double logn_auto_b  = logn_auto.val_grid_point(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 ) ;
          
        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_c  = logn_auto.val_grid_point(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.set_spectral_va().smooth(nzet, logn_auto.set_spectral_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 ;
    cout.precision(8) ;
    cout << "pot" << norme(pot_ext) << endl ;

    (ent.set_spectral_va()).smooth(nzet, ent.set_spectral_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
    }

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

    Map_et mp_prev = mp_et ; 

    Cmp ent_cmp(ent) ;
    mp.adapt(ent_cmp, par_adapt) ; 
    ent = ent_cmp ;

    // Readjustment of the external boundary of domain l=nzet
    // to keep a fixed ratio with respect to star's surface
    
    if (nz>= 5) {
      double separation = 2. * fabs(mp.get_ori_x()) ;
      double ray_eqq = ray_eq() ;
      double ray_eqq_pi = ray_eq_pi() ;
      double new_rr_out_2 = (separation - ray_eqq) * 0.95 ; 
      double new_rr_out_3 = (separation + ray_eqq_pi) * 1.05 ; 
      
      double rr_in_1 = mp.val_r(1,-1., M_PI/2, 0.) ; 
      double rr_out_1 = mp.val_r(1, 1., M_PI/2, 0.) ; 
      double rr_out_2 = mp.val_r(2, 1., M_PI/2, 0.) ; 
      double rr_out_3 = mp.val_r(3, 1., M_PI/2, 0.) ; 
      
      mp.resize(1, 0.5*(new_rr_out_2 + rr_in_1) / rr_out_1) ; 
      mp.resize(2, new_rr_out_2 / rr_out_2) ; 
      mp.resize(3, new_rr_out_3 / rr_out_3) ;

      for (int dd=4; dd<=nz-2; dd++) {
        mp.resize(dd, new_rr_out_3 * pow(4., dd-3) / 
              mp.val_r(dd, 1., M_PI/2, 0.)) ;
      }

    }
    else {
      cout << "too small number of domains" << endl ;
    }
    
    //----------------------------------------------------
    // Computation of the enthalpy at the new grid points
    //----------------------------------------------------
    
    mp_prev.homothetie(alpha_r) ; 
    
    Cmp ent_cmp2 (ent) ;
    mp.reevaluate_symy(&mp_prev, nzet+1, ent_cmp2) ; 
    ent = ent_cmp2 ;

    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.val_grid_point(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)

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

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

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

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

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

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

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

    const Scalar& divbeta_auto = beta_auto.divergence(flat) ;
    const Tensor& dcov_beta_auto = beta_auto.derive_cov(flat) ;
    Tensor dcovdcov_beta_auto = beta_auto.derive_cov(flat)
      .derive_cov(flat) ;
    dcovdcov_beta_auto.inc_dzpuis() ;


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

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

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

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

    Connection gamijk (gtilde, flat) ;
    const Tensor& deltaijk = gamijk.get_delta() ;

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

    double lambda_dirac = 0. ;

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


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

    int nr = mp.get_mg()->get_nr(0) ;
    int nt = mp.get_mg()->get_nt(0) ;
    int np = mp.get_mg()->get_np(0) ;
    
    Scalar source1(mp) ;
    Scalar source2(mp) ;
    Scalar source3(mp) ;
    Scalar source4(mp) ;
    Scalar source5(mp) ;
    Scalar source6(mp) ;
    Scalar source7(mp) ;
    Scalar source8(mp) ;
     
    source1 = qpig * psi4 % (ener_euler + s_euler) ; 

    source2 = psi4 % (kcar_auto + kcar_comp) ;

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

    source5 = - contract(hdirac, 0, dcov_logn_auto, 0) ;

    source_tot = source1 + source2 + source3 + source4 + source5 ;
      
 
    cout << "moyenne de la source 1 pour logn_auto" << endl ;
    cout <<  norme(source1/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 2 pour logn_auto" << endl ;
    cout <<  norme(source2/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 3 pour logn_auto" << endl ;
    cout <<  norme(source3/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 4 pour logn_auto" << endl ;
    cout <<  norme(source4/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 5 pour logn_auto" << endl ;
    cout <<  norme(source5/(nr*nt*np)) << endl ;
    cout << "moyenne de la source pour logn_auto" << endl ;
    cout <<  norme(source_tot/(nr*nt*np)) << endl ;

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

    source_tot.poisson(par_logn, logn_auto) ; 
    ssjm1_logn = ssjm1logn ;

    cout << "logn_auto" << endl << norme(logn_auto/(nr*nt*np)) << endl ;
  
    // Check: has the Poisson equation been correctly solved ?
    // -----------------------------------------------------

    Tbl tdiff_logn = diffrel(logn_auto.laplacian(), source_tot) ;
    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)) ; 

    //--------------------------------------------------------
    // Poisson equation for lnq_auto
    //--------------------------------------------------------

    // Source
    //--------

    source1 = qpig * psi4 % s_euler ;

    source2 = 0.75 * psi4 % (kcar_auto + kcar_comp) ;

    source3 = - contract(dcon_lnq, 0, dcov_lnq_auto, 0, true) ;

    source4 = 2. * contract(contract(gtilde_con, 0, dcov_phi, 0), 0, 
                dcov_phi_auto + dcov_logn_auto, 0, true) ;

    source5 = 0.0625 * contract(gtilde_con, 0, 1, contract(
         dcov_hij_auto, 0, 1, dcov_gtilde, 0, 1), 0, 1) ;
        
    source6 = - 0.125 * contract(gtilde_con, 0, 1, contract(
               dcov_hij_auto, 0, 1, dcov_gtilde, 0, 2), 0, 1) ;

    source7 = - contract(hij, 0, 1, dcovdcov_lnq_auto + dcov_lnq_auto *
                 dcov_lnq, 0, 1) ;

    source8 = - 0.25 * contract(dcov_hdirac_auto, 0, 1) 
      - contract(hdirac, 0, dcov_lnq_auto, 0) ;
    
    source_tot = source1 + source2 + source3 + source4 + source5 + 
                 source6 + source7 + source8 ;

    
    cout << "moyenne de la source 1 pour lnq_auto" << endl ;
    cout <<  norme(source1/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 2 pour lnq_auto" << endl ;
    cout <<  norme(source2/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 3 pour lnq_auto" << endl ;
    cout <<  norme(source3/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 4 pour lnq_auto" << endl ;
    cout <<  norme(source4/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 5 pour lnq_auto" << endl ;
    cout <<  norme(source5/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 6 pour lnq_auto" << endl ;
    cout <<  norme(source6/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 7 pour lnq_auto" << endl ;
    cout <<  norme(source7/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 8 pour lnq_auto" << endl ;
    cout <<  norme(source8/(nr*nt*np)) << endl ;
    cout << "moyenne de la source pour lnq_auto" << endl ;
    cout <<  norme(source_tot/(nr*nt*np)) << endl ;
    

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

    source_tot.poisson(par_lnq, lnq_auto) ; 
    ssjm1_lnq = ssjm1lnq ;

    cout << "lnq_auto" << endl << norme(lnq_auto/(nr*nt*np)) << endl ;

    // Check: has the Poisson equation been correctly solved 
    // -----------------------------------------------------
    
    Tbl tdiff_lnq = diffrel(lnq_auto.laplacian(), source_tot) ;
    cout << 
        "Relative error in the resolution of the equation for lnq : "
         << endl ; 
    for (int l=0; l<nz; l++) {
        cout << tdiff_lnq(l) << "  " ; 
    }
    cout << endl ;
    diff_lnq = max(abs(tdiff_lnq)) ; 

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

    // Source
    //--------

    Vector source1_beta(mp, CON, mp.get_bvect_cart()) ;
    Vector source2_beta(mp, CON, mp.get_bvect_cart()) ;
    Vector source3_beta(mp, CON, mp.get_bvect_cart()) ;
    Vector source4_beta(mp, CON, mp.get_bvect_cart()) ;
    Vector source5_beta(mp, CON, mp.get_bvect_cart()) ;
    Vector source6_beta(mp, CON, mp.get_bvect_cart()) ;
    Vector source7_beta(mp, CON, mp.get_bvect_cart()) ;

    source1_beta = (4.*qpig) * nn % psi4
                       %(ener_euler + press) * u_euler ;

    source2_beta = 2. * nn * contract(tkij_auto, 1, 
                        dcov_logn - 6 * dcov_phi, 0)  ;

    source3_beta = - 2. * nn * contract(tkij_auto, 0, 1, deltaijk, 
                          1, 2) ;

    source4_beta = - contract(hij, 0, 1, dcovdcov_beta_auto, 1, 2) ;
    
    source5_beta = - 0.3333333333333333 * contract(hij, 1, contract( 
                          dcovdcov_beta_auto, 0, 1), 0) ;
    
    source6_beta.set_etat_zero() ; //hdirac_auto.derive_lie(omdsdp) ;
    source6_beta.inc_dzpuis() ;

    source7_beta = contract(beta, 0, dcov_hdirac_auto, 1) ;
           + 2./3. * hdirac * divbeta_auto 
           - contract(hdirac, 0, dcov_beta_auto, 1) ;

    source_beta = source1_beta + source2_beta + source3_beta 
      + source4_beta + source5_beta + source6_beta + source7_beta ; 
    

    cout << "moyenne de la source 1 pour beta_auto" << endl ;
    cout <<  norme(source1_beta(1)/(nr*nt*np)) << endl ;
    cout <<  norme(source1_beta(2)/(nr*nt*np)) << endl ;
    cout <<  norme(source1_beta(3)/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 2 pour beta_auto" << endl ;
    cout <<  norme(source2_beta(1)/(nr*nt*np)) << endl ;
    cout <<  norme(source2_beta(2)/(nr*nt*np)) << endl ;
    cout <<  norme(source2_beta(3)/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 3 pour beta_auto" << endl ;
    cout <<  norme(source3_beta(1)/(nr*nt*np)) << endl ;
    cout <<  norme(source3_beta(2)/(nr*nt*np)) << endl ;
    cout <<  norme(source3_beta(3)/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 4 pour beta_auto" << endl ;
    cout <<  norme(source4_beta(1)/(nr*nt*np)) << endl ;
    cout <<  norme(source4_beta(2)/(nr*nt*np)) << endl ;
    cout <<  norme(source4_beta(3)/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 5 pour beta_auto" << endl ;
    cout <<  norme(source5_beta(1)/(nr*nt*np)) << endl ;
    cout <<  norme(source5_beta(2)/(nr*nt*np)) << endl ;
    cout <<  norme(source5_beta(3)/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 6 pour beta_auto" << endl ;
    cout <<  norme(source6_beta(1)/(nr*nt*np)) << endl ;
    cout <<  norme(source6_beta(2)/(nr*nt*np)) << endl ;
    cout <<  norme(source6_beta(3)/(nr*nt*np)) << endl ;
    cout << "moyenne de la source 7 pour beta_auto" << endl ;
    cout <<  norme(source7_beta(1)/(nr*nt*np)) << endl ;
    cout <<  norme(source7_beta(2)/(nr*nt*np)) << endl ;
    cout <<  norme(source7_beta(3)/(nr*nt*np)) << endl ;
    cout << "moyenne de la source pour beta_auto" << endl ;
    cout <<  norme(source_beta(1)/(nr*nt*np)) << endl ;
    cout <<  norme(source_beta(2)/(nr*nt*np)) << endl ;
    cout <<  norme(source_beta(3)/(nr*nt*np)) << endl ;

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

    // Filter for the source of beta vector
    
    for (int i=1; i<=3; i++) {
      if (source_beta(i).get_etat() != ETATZERO)
        source_beta.set(i).filtre(4) ;
    }
    
    for (int i=1; i<=3; i++) {
      if(source_beta(i).dz_nonzero()) {
        assert( source_beta(i).get_dzpuis() == 4 ) ; 
      }
      else{
        (source_beta.set(i)).set_dzpuis(4) ; 
      }
    }
    
    double lambda = double(1) / double(3) ; 

    Tenseur source_p(mp, 1, CON, mp.get_bvect_cart() ) ;
    source_p.set_etat_qcq() ;
    for (int i=0; i<3; i++) {
      source_p.set(i) = Cmp(source_beta(i+1)) ;
    }
    
    Tenseur vect_auxi (mp, 1, CON, mp.get_bvect_cart()) ;
    vect_auxi.set_etat_qcq() ;
    for (int i=0; i<3 ;i++){
      vect_auxi.set(i) = 0. ;
    }
    Tenseur scal_auxi (mp) ;
    scal_auxi.set_etat_qcq() ;
    scal_auxi.set().annule_hard() ;
    scal_auxi.set_std_base() ;
    
    Tenseur resu_p(mp, 1, CON, mp.get_bvect_cart() ) ;
    resu_p.set_etat_qcq() ;
    for (int i=0; i<3 ;i++)
      resu_p.set(i).annule_hard() ;
    resu_p.set_std_base() ;

    //source_p.poisson_vect(lambda, par_beta2, resu_p, vect_auxi, 
    //            scal_auxi) ;
    
    source_p.poisson_vect_oohara(lambda, par_beta2, resu_p, scal_auxi) ;

    for (int i=1; i<=3; i++) 
      beta_auto.set(i) = resu_p(i-1) ;

    ssjm1_khi = ssjm1khi ;
    for (int i=0; i<3; i++){
        ssjm1_wbeta.set(i+1) = ssjm1wbeta(i) ;
    }
    
    cout << "beta_auto_x" << endl << norme(beta_auto(1)/(nr*nt*np)) 
         << endl ;
    cout << "beta_auto_y" << endl << norme(beta_auto(2)/(nr*nt*np)) 
         << endl ;
    cout << "beta_auto_z" << endl << norme(beta_auto(3)/(nr*nt*np)) 
         << endl ;
  

    // Check: has the equation for beta_auto been correctly solved ?
    // --------------------------------------------------------------
    
    Vector lap_beta = (beta_auto.derive_con(flat)).divergence(flat) 
        + lambda* beta_auto.divergence(flat).derive_con(flat) ;
    
    source_beta.dec_dzpuis() ;
    Tbl tdiff_beta_x = diffrel(lap_beta(1), source_beta(1)) ; 
    Tbl tdiff_beta_y = diffrel(lap_beta(2), source_beta(2)) ; 
    Tbl tdiff_beta_z = diffrel(lap_beta(3), source_beta(3)) ; 

    cout << 
        "Relative error in the resolution of the equation for beta_auto : "
         << endl ; 
    cout << "x component : " ;
    for (int l=0; l<nz; l++) {
        cout << tdiff_beta_x(l) << "  " ; 
    }
    cout << endl ;
    cout << "y component : " ;
    for (int l=0; l<nz; l++) {
        cout << tdiff_beta_y(l) << "  " ; 
    }
    cout << endl ;
    cout << "z component : " ;
    for (int l=0; l<nz; l++) {
        cout << tdiff_beta_z(l) << "  " ; 
    }
    cout << endl ;
    
    diff_beta_x = max(abs(tdiff_beta_x)) ; 
    diff_beta_y = max(abs(tdiff_beta_y)) ; 
    diff_beta_z = max(abs(tdiff_beta_z)) ; 


    if (!conf_flat) {
       
        //--------------------------------------------------------
        // Poisson equation for hij
        //--------------------------------------------------------
     
     
        // Declaration of all sources 
        //---------------------------

        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()) ;

        // Sources
        //-----------

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

        // Function exp(-(r-r_0)^2/sigma^2)
        // --------------------------------
        
        double r0 = mp.val_r(nz-2, 1, 0, 0) ;
        double sigma = 1.*r0 ;
        
        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() ;
        
        //  ff.annule_domain(nz-1) ;
        //des_profile(ff, 0, 20, 0, 0) ;

        // 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_auto, 0, dcov_omdsdphi, 1) ;
        source_3a.inc_dzpuis() ;

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

        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() ;

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

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

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

        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, dcov_phi, 0) ;

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

        tens_temp = nn / (3.*psi4) * hdirac.divergence(flat)*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, dcov_logn, 0), 0) * gtilde_con ;

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

        source_Sij += - 2. * qpig * nn * ( psi4 * stress_euler 
                  - 0.33333333333333333 * 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(hij, 1,
                    dcov_hij_auto, 2), 1, dcov_qq, 0) -
          0.5/(psi4*psi2) * contract(contract(dcov_hij_auto, 2,
                          hij, 1), 1, dcov_qq, 0) ;
                    
        source_Sij += 0.5/(psi4*psi2) * contract(contract(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)*hij_auto ;

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

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


        source_Rij += - contract(hij_auto, 1, dcov_hdirac, 1) -
          contract(dcov_hdirac, 1, 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 ;

            //source_tot_hij.inc_dzpuis() ;

            cout << "source_mat" << endl 
             << norme((- 2. * qpig * nn * ( psi4 * stress_euler 
                   - 0.33333333333333333 * s_euler * gtilde_con ))
                  (i,j))/(nr*nt*np) << endl ;
            cout << "max source_mat" << endl 
             << max((- 2. * qpig * nn * ( psi4 * stress_euler 
                   - 0.33333333333333333 * s_euler * gtilde_con ))
                  (i,j)) << endl ;
        
            cout << "source1" << endl 
             << norme(source_1(i,j)/(nr*nt*np)) << endl ;
            cout << "max source1" << endl 
             << max(source_1(i,j)) << endl ;
            cout << "source2" << endl 
             << norme(source_2(i,j)/(nr*nt*np)) << endl ;
            cout << "max source2" << endl 
             << max(source_2(i,j)) << endl ;
            cout << "source3a" << endl 
             << norme(source_3a(i,j)/(nr*nt*np)) << endl ;
            cout << "max source3a" << endl 
             << max(source_3a(i,j)) << endl ;
                    cout << "source3b" << endl
                         << norme(source_3b(i,j)/(nr*nt*np)) << endl ;
                    cout << "max source3b" << endl
                         << max(source_3b(i,j)) << endl ;
            cout << "source4" << endl 
             << norme(source_4(i,j)/(nr*nt*np)) << endl ;
            cout << "max source4" << endl 
             << max(source_4(i,j)) << endl ;
            cout << "source5" << endl 
             << norme(source_5(i,j)/(nr*nt*np)) << endl ;
            cout << "max source5" << endl 
             << max(source_5(i,j)) << endl ;
            cout << "source6" << endl 
             << norme(source_6(i,j)/(nr*nt*np)) << endl ;
            cout << "max source6" << endl 
             << max(source_6(i,j)) << endl ;
            cout << "source_Rij" << endl 
             << norme(source_Rij(i,j)/(nr*nt*np)) << endl ;
            cout << "max source_Rij" << endl 
             << max(source_Rij(i,j)) << endl ;
            cout << "source_Sij" << endl 
             << norme(source_Sij(i,j)/(nr*nt*np)) << endl ;
            cout << "max source_Sij" << endl 
             << max(source_Sij(i,j)) << endl ;
            cout << "source_tot" << endl 
             << norme(source_tot_hij/(nr*nt*np)) << endl ;
            cout << "max source_tot" << endl 
             << max(source_tot_hij) << endl ;
                
            // Resolution of the Poisson equations and
            // Check: has the Poisson equation been correctly solved ?
            // -----------------------------------------------------

            if(i==1 && j==1) {
         
            source_tot_hij.poisson(par_h11, hij_auto.set(1,1)) ; 
            
            Scalar laplac (hij_auto(1,1).laplacian()) ;
            laplac.dec_dzpuis() ;
            Tbl tdiff_h11 = diffrel(laplac, source_tot_hij) ;  
            cout << "Relative error in the resolution of the equation for "
                 << "h11_auto : " << endl ; 
            for (int l=0; l<nz; l++) {
                cout << tdiff_h11(l) << "  " ; 
            }
            cout << endl ;
            diff_h11 = max(abs(tdiff_h11)) ; 
            }
                   
            if(i==2 && j==1) {

            source_tot_hij.poisson(par_h21, hij_auto.set(2,1)) ; 
        
            Scalar laplac (hij_auto(2,1).laplacian()) ;
            laplac.dec_dzpuis() ;
            Tbl tdiff_h21 = diffrel(laplac, source_tot_hij) ;  
    
            cout << 
                "Relative error in the resolution of the equation for " 
                 << "h21_auto : "  << endl ; 
            for (int l=0; l<nz; l++) {
                cout << tdiff_h21(l) << "  " ; 
            }
            cout << endl ;
            diff_h21 = max(abs(tdiff_h21)) ; 
            }
           
            if(i==3 && j==1) {
         
            source_tot_hij.poisson(par_h31, hij_auto.set(3,1)) ; 

            Scalar laplac (hij_auto(3,1).laplacian()) ;
            laplac.dec_dzpuis() ;
            Tbl tdiff_h31 = diffrel(laplac, source_tot_hij) ;  

            cout << 
                "Relative error in the resolution of the equation for "
                 << "h31_auto : " << endl ; 
            for (int l=0; l<nz; l++) {
                cout << tdiff_h31(l) << "  " ; 
            }
            cout << endl ;
            diff_h31 = max(abs(tdiff_h31)) ; 
            }
         
            if(i==2 && j==2) {
         
            source_tot_hij.poisson(par_h22, hij_auto.set(2,2)) ; 

            Scalar laplac (hij_auto(2,2).laplacian()) ;
            laplac.dec_dzpuis() ;
            Tbl tdiff_h22 = diffrel(laplac, source_tot_hij) ;  

            cout << 
                "Relative error in the resolution of the equation for "
                 << "h22_auto : " << endl ; 
            for (int l=0; l<nz; l++) {
                cout << tdiff_h22(l) << "  " ; 
            }
            cout << endl ;
            diff_h22 = max(abs(tdiff_h22)) ; 
            }
           
            if(i==3 && j==2) {
         
            source_tot_hij.poisson(par_h32, hij_auto.set(3,2)) ; 
    
            Scalar laplac (hij_auto(3,2).laplacian()) ;
            laplac.dec_dzpuis() ;
            Tbl tdiff_h32 = diffrel(laplac, source_tot_hij) ;  

            cout << 
                "Relative error in the resolution of the equation for "
                 << "h32_auto : " << endl ; 
            for (int l=0; l<nz; l++) {
                cout << tdiff_h32(l) << "  " ; 
            }
            cout << endl ;
            diff_h32 = max(abs(tdiff_h32)) ; 
            }
         
            if(i==3 && j==3) {
         
            source_tot_hij.poisson(par_h33, hij_auto.set(3,3)) ; 

            Scalar laplac (hij_auto(3,3).laplacian()) ;
            laplac.dec_dzpuis() ;
            Tbl tdiff_h33 = diffrel(laplac, source_tot_hij) ;  

            cout << 
                "Relative error in the resolution of the equation for "
                 << "h33_auto : " << endl ;
            for (int l=0; l<nz; l++) {
                cout << tdiff_h33(l) << "  " ;
            }
            cout << endl ;
            diff_h33 = max(abs(tdiff_h33)) ;
            }

        }
      
        cout << "Tenseur hij auto in cartesian coordinates" << endl ;
        for (int i=1; i<=3; i++)
        for (int j=1; j<=i; j++) {
            cout << "  Comp. " << i << " " << j << " :  " ;
            for (int l=0; l<nz; l++){
            cout << norme(hij_auto(i,j)/(nr*nt*np))(l) << " " ;
            }
            cout << endl ;
        }
        cout << endl ;

        ssjm1_h11 = ssjm1h11 ;
        ssjm1_h21 = ssjm1h21 ;
        ssjm1_h31 = ssjm1h31 ;
        ssjm1_h22 = ssjm1h22 ;
        ssjm1_h32 = ssjm1h32 ;
        ssjm1_h33 = ssjm1h33 ;

    }

    // End of relativistic equations    
       
           
    //-------------------------------------------------
    //  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
    //=========================================================================

}  





    
    
}