et_bin_equil_regu.C

/*
 *  Method of class Etoile_bin to compute an equilibrium configuration
 *  by regularizing source.
 *
 *  (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
 *
 */
 
 
 
 
/*
 * $Id: et_bin_equil_regu.C,v 1.9 2016/12/05 16:17:52 j_novak Exp $
 * $Log: et_bin_equil_regu.C,v $
 * Revision 1.9  2016/12/05 16:17:52  j_novak
 * Suppression of some global variables (file names, loch, ...) to prevent redefinitions
 *
 * Revision 1.8  2014/10/13 08:52:55  j_novak
 * Lorene classes and functions now belong to the namespace Lorene.
 *
 * Revision 1.7  2014/10/06 15:13:08  j_novak
 * Modified #include directives to use c++ syntax.
 *
 * Revision 1.6  2009/06/15 09:26:57  k_taniguchi
 * Improved the rescaling of the domains.
 *
 * Revision 1.5  2004/03/25 10:29:03  j_novak
 * All LORENE's units are now defined in the namespace Unites (in file unites.h).
 *
 * Revision 1.4  2003/09/01 06:48:08  k_taniguchi
 * Change of the domain which should be resized.
 *
 * Revision 1.3  2003/08/31 05:35:38  k_taniguchi
 * Addition of the specification of the domain
 *  which is resized.
 *
 * Revision 1.2  2002/12/11 12:51:26  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 2.17  2001/08/07  09:49:00  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::equil_regular : Tbl fact.
 *
 * Revision 2.16  2001/06/22  08:54:53  keisuke
 * Set the inner values of the second domain of ent
 *   by using the outer ones of the first domain.
 *
 * Revision 2.15  2001/05/17  12:22:26  keisuke
 * Change of the method to calculate chi from setting position in map
 *  to val_point.
 *
 * Revision 2.14  2001/02/07  09:47:28  eric
 * unsgam1 est desormais donne par Eos::der_nbar_ent (cas newtonien)
 *   ou Eos::der_ener_ent (cas relativiste).
 *
 * Revision 2.13  2001/01/16  17:02:32  keisuke
 * *** empty log message ***
 *
 * Revision 2.12  2001/01/16  16:58:08  keisuke
 * Change the method to set the values on the surface.
 *
 * Revision 2.11  2001/01/10  16:45:34  keisuke
 * Set the inner values of the second domain of logn_auto
 *   by using the outer ones of the first domain.
 *
 * Revision 2.10  2000/12/20  10:33:14  eric
 * Changement important : nz_search = nzet ---> nz_search = nzet + 1
 *
 * Revision 2.9  2000/10/25  14:01:03  keisuke
 * Modif de Map_et::adapt: on y rentre desormais avec nz_search
 *  (dans le cas present nz_search = nzet).
 *
 * Revision 2.8  2000/10/06  15:29:01  keisuke
 * Change poisson_vect into poisson_vect_regu.
 *
 * Revision 2.7  2000/09/25  15:01:10  keisuke
 * Suppress "int" from the declaration of k_div.
 *
 * Revision 2.6  2000/09/22  15:51:39  keisuke
 * d_logn_auto est desormais calcule en dehors (dans update_metric).
 *
 * Revision 2.5  2000/09/13  09:50:33  keisuke
 * Minor change on change_triad.
 *
 * Revision 2.4  2000/09/08  15:57:31  keisuke
 * Change the basis of d_logn_auto_div from the spherical coordinate
 *  to the Cartesian one with respect to ref_triad.
 *
 * Revision 2.3  2000/09/07  15:47:19  keisuke
 * Minor change.
 *
 * Revision 2.2  2000/09/07  15:43:41  keisuke
 * Add a new argument in poisson_regular and suppress logn_auto_total.
 *
 * Revision 2.1  2000/08/29  14:01:43  keisuke
 * Modify the arguments of poisson_regular.
 *
 * Revision 2.0  2000/08/29  11:39:02  eric
 * Version provisoire.
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Etoile/et_bin_equil_regu.C,v 1.9 2016/12/05 16:17:52 j_novak Exp $
 *
 */
 
// Headers C
#include <cmath>
 
// Headers Lorene
#include "etoile.h"
#include "param.h"
#include "eos.h"
#include "utilitaires.h"
#include "unites.h"     
 
namespace Lorene {
 
//********************************************************************
 
void Etoile_bin::equil_regular(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) {
 
    // Fundamental constants and units
    // -------------------------------
  using namespace Unites ;
 
    // Initializations
    // ---------------
 
    k_div = 2 ;  // Regularity parameter for poisson_regular
 
    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
 
    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 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 
    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_regular 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_regu
    // ------------------------------------------------------
 
    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
    // ------------------
 
    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
 
    Cmp source_regu(mp) ;
    Cmp source_div(mp) ;
 
    //=========================================================================
    //          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) ; 
 
        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
    //--------------------
 
    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
    }
 
 
    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 ;
 
//##
//  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
 
//##
//  des_coupe_z(ent(), 0., 1, "ent after adapt", &(ent()) ) ;
//##
    //----------------------------------------------------
    // 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
    // ------
 
    double unsgam1 ;    // effective power of H in the source 
                // close to the surface
 
    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) ;
                       
        // 1/(gam-1) = dln(e)/dln(H) close to the surface : 
        unsgam1 = eos.der_ener_ent_p(ent_b + 1e-10*(ent_c-ent_b)) ; 
    
    }
    else {
        source = qpig * nbar ;
        
        // 1/(gam-1) = dln(n)/dln(H) close to the surface : 
        unsgam1 = eos.der_nbar_ent_p(ent_b + 1e-10*(ent_c-ent_b)) ; 
    }
 
    source.set_std_base() ;
 
    // Resolution of the Poisson equation
    // ----------------------------------
 
    logn_auto_regu.set_etat_qcq() ;
    logn_auto_div.set_etat_qcq() ;
    d_logn_auto_div.set_etat_qcq() ;
 
    source_regu.std_base_scal() ;
    source_div.std_base_scal() ;
 
    source().poisson_regular(k_div, nzet, unsgam1, par_poisson1,
                 logn_auto.set(), logn_auto_regu.set(),
                 logn_auto_div.set(),
                 d_logn_auto_div,
                 source_regu, source_div) ;
 
    // 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
        // ------
 
        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
        // ------
 
        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_regu,
        // 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_regu(k_div, nzet, unsgam1,
                       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
 
 
    //-------------------------------------------------
    //  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
    //=========================================================================
 
 
}
 
}