et_bin_bhns_extr_equil.C

/*
 *  Method of class Et_bin_bhns_extr to compute an equilibrium configuration
 *  of a BH-NS binary system with an extreme mass ratio
 *
 *    (see file et_bin_bhns_extr.h for documentation).
 *
 */

/*
 *   Copyright (c) 2004-2005 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 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
 *
 */

char et_bin_bhns_extr_equil_C[] = "$Header: /cvsroot/Lorene/C++/Source/Etoile/et_bin_bhns_extr_equil.C,v 1.9 2005/02/28 23:11:05 k_taniguchi Exp $" ;

/*
 * $Id: et_bin_bhns_extr_equil.C,v 1.9 2005/02/28 23:11:05 k_taniguchi Exp $
 * $Log: et_bin_bhns_extr_equil.C,v $
 * Revision 1.9  2005/02/28 23:11:05  k_taniguchi
 * Modification to include the case of the conformally flat background metric
 *
 * Revision 1.8  2005/01/25 17:33:19  k_taniguchi
 * Suppression of the filter for the source term of the shift vector.
 *
 * Revision 1.7  2005/01/03 18:01:12  k_taniguchi
 * Addition of the method to fix the position of the neutron star
 * in the coordinate system.
 *
 * Revision 1.6  2004/12/29 16:29:55  k_taniguchi
 * Suppression of "dzpius" for the shift vector.
 *
 * Revision 1.5  2004/12/22 18:26:53  k_taniguchi
 * Change an argument of poisson_vect_falloff.
 *
 * Revision 1.4  2004/12/06 17:59:50  k_taniguchi
 * Change the position of resize.
 *
 * Revision 1.3  2004/12/02 21:31:56  k_taniguchi
 * Set a filter for the shift vector.
 *
 * Revision 1.2  2004/12/02 15:05:36  k_taniguchi
 * Modification of the procedure for resize.
 *
 * Revision 1.1  2004/11/30 20:48:45  k_taniguchi
 * *** empty log message ***
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Etoile/et_bin_bhns_extr_equil.C,v 1.9 2005/02/28 23:11:05 k_taniguchi Exp $
 *
 */

// C headers
#include <math.h>

// Lorene headers
#include "et_bin_bhns_extr.h"
#include "etoile.h"
#include "map.h"
#include "coord.h"
#include "param.h"
#include "eos.h"
#include "graphique.h"
#include "utilitaires.h"
#include "unites.h"

void Et_bin_bhns_extr::equil_bhns_extr_ks(double ent_c, const double& mass,
                      const double& sepa, int mermax,
                      int mermax_poisson,
                      double relax_poisson,
                      int mermax_potvit,
                      double relax_potvit, int np_filter,
                      double thres_adapt,
                      Tbl& diff) {

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

    assert( kerrschild == true ) ;

    // 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 ;  // 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 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 ;

    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_pot_extr(mass, sepa, 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 ) ;

        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 ;

    //----------------------------------------------------------
    // Change the enthalpy field to be set its maximum position
    // at the coordinate center
    //----------------------------------------------------------

    double dentdx = ent().dsdx()(0, 0, 0, 0) ;
    double dentdy = ent().dsdy()(0, 0, 0, 0) ;

    cout << "dH/dx|_center = " << dentdx << endl ;
    cout << "dH/dy|_center = " << dentdy << endl ;

    double dec_fact = 1. ;

    Tenseur func_in(mp) ;
    func_in.set_etat_qcq() ;
    func_in.set() = 1. - dec_fact * (dentdx/ent_c) * mp.x
      - dec_fact * (dentdy/ent_c) * mp.y ;
    func_in.set().annule(nzet, nz-1) ;
    func_in.set_std_base() ;

    Tenseur func_ex(mp) ;
    func_ex.set_etat_qcq() ;
    func_ex.set() = 1. ;
    func_ex.set().annule(0, nzet-1) ;
    func_ex.set_std_base() ;

    // New enthalpy field
    // ------------------
    ent.set() = ent() * (func_in() + func_ex()) ;

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

    double dentdx_new = ent().dsdx()(0, 0, 0, 0) ;
    double dentdy_new = ent().dsdy()(0, 0, 0, 0) ;
    cout << "dH/dx|_new    = " << dentdx_new << endl ;
    cout << "dH/dy|_new    = " << dentdy_new << endl ;

    //-----------------------------------------------------
    // Adaptation of the mapping to the new enthalpy field
    //----------------------------------------------------

    // Shall the adaptation be performed (cusp) ?
    // ------------------------------------------

    double dent_eq = ent().dsdr().val_point(ray_eq_pi(),M_PI/2.,M_PI) ;
    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 ;

    mp.adapt(ent(), par_adapt) ;

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

    mp_prev.homothetie(alpha_r) ;

    mp.reevaluate(&mp_prev, nzet+1, ent.set()) ;

    double fact_resize = 1. / alpha_r ;
    for (int l=nzet; l<nz-1; l++) {
        mp_et.resize(l, fact_resize) ;
    }
    mp_et.resize_extr(fact_resize) ;

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


    //----------------------------------------------------
    // Auxiliary terms for the source of Poisson equation
    //----------------------------------------------------

    const Coord& xx = mp.x ;
    const Coord& yy = mp.y ;
    const Coord& zz = mp.z ;

    Tenseur r_bh(mp) ;
    r_bh.set_etat_qcq() ;
    r_bh.set() = pow( (xx+sepa)*(xx+sepa) + yy*yy + zz*zz, 0.5) ;
    r_bh.set_std_base() ;

    Tenseur xx_cov(mp, 1, COV, ref_triad) ;
    xx_cov.set_etat_qcq() ;
    xx_cov.set(0) = xx + sepa ;
    xx_cov.set(1) = yy ;
    xx_cov.set(2) = zz ;
    xx_cov.set_std_base() ;

    Tenseur xsr_cov(mp, 1, COV, ref_triad) ;
    xsr_cov = xx_cov / r_bh ;
    xsr_cov.set_std_base() ;

    Tenseur msr(mp) ;
    msr = ggrav * mass / r_bh ;
    msr.set_std_base() ;

    Tenseur lapse_bh(mp) ;
    lapse_bh = 1. / sqrt( 1.+2.*msr ) ;
    lapse_bh.set_std_base() ;

    Tenseur lapse_bh2(mp) ;  // lapse_bh * lapse_bh
    lapse_bh2 = 1. / (1.+2.*msr) ;
    lapse_bh2.set_std_base() ;

    Tenseur ldnu(mp) ;
    ldnu = flat_scalar_prod_desal(xsr_cov, d_logn_auto) ;
    ldnu.set_std_base() ;

    Tenseur ldbeta(mp) ;
    ldbeta = flat_scalar_prod_desal(xsr_cov, d_beta_auto) ;
    ldbeta.set_std_base() ;

    Tenseur lltkij(mp) ;
    lltkij.set_etat_qcq() ;
    lltkij.set() = 0 ;
    lltkij.set_std_base() ;

    for (int i=0; i<3; i++)
        for (int j=0; j<3; j++)
            lltkij.set() += xsr_cov(i) * xsr_cov(j) * tkij_auto(i, j) ;

    Tenseur lshift(mp) ;
    lshift = flat_scalar_prod_desal(xsr_cov, shift_auto) ;
    lshift.set_std_base() ;

    Tenseur d_ldnu(mp, 1, COV, ref_triad) ;
    d_ldnu = ldnu.gradient() ;        // (d/dx, d/dy, d/dz)
    d_ldnu.change_triad(ref_triad) ;  // --> (d/dX, d/dY, d/dZ)

    Tenseur ldldnu(mp) ;
    ldldnu = flat_scalar_prod_desal(xsr_cov, d_ldnu) ;
    ldldnu.set_std_base() ;

    Tenseur d_ldbeta(mp, 1, COV, ref_triad) ;
    d_ldbeta = ldbeta.gradient() ;      // (d/dx, d/dy, d/dz)
    d_ldbeta.change_triad(ref_triad) ;  // --> (d/dX, d/dY, d/dZ)

    Tenseur ldldbeta(mp) ;
    ldldbeta = flat_scalar_prod_desal(xsr_cov, d_ldbeta) ;
    ldldbeta.set_std_base() ;

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

    // Source
    // ------

    if (relativistic) {

        source = qpig * a_car % (ener_euler + s_euler) + akcar_auto
          - flat_scalar_prod_desal(d_logn_auto, d_beta_auto)
          + 2.*lapse_bh2 % msr % (ldnu % ldbeta + ldldnu)
          + lapse_bh2 % lapse_bh2 % msr % (2.*(ldnu + 4.*msr % ldnu)
                           - ldbeta) / r_bh
          - (4.*a_car % lapse_bh2 % lapse_bh2 % msr / 3. / nnn / r_bh)
          * (2.+3.*msr) * (3.+4.*msr) * lltkij
          + (2.*a_car % lapse_bh2 % lapse_bh2 % lapse_bh % msr
         / nnn / r_bh / r_bh) * (2.+10.*msr+9.*msr%msr) * lshift
          + (4.*pow(lapse_bh2, 3.) % msr % msr / 3. / r_bh / r_bh)
          % (2.*(a_car%lapse_bh2/nnn/nnn - 1.) * pow(2.+3.*msr, 2.)
         + (a_car - 1.) % pow(1.+3.*msr, 2.)
         - 3.*(a_car%lapse_bh/nnn - 1.)*(2.+10.*msr+9.*msr%msr)) ;

    }
    else {
        cout << "The computation of BH-NS binary systems"
         << " should be relativistic !!!" << endl ;
        abort() ;
    }

    source.set_std_base() ;

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

    int k_falloff = 1 ;

    source().poisson_falloff(par_poisson1, logn_auto.set(), k_falloff) ;

    // Construct logn_auto_regu for et_bin_upmetr_extr.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
        // ------

        source = qpig * a_car % s_euler + 0.75 * akcar_auto
          - 0.5 * flat_scalar_prod_desal(d_logn_auto, d_logn_auto)
          - 0.5 * flat_scalar_prod_desal(d_beta_auto, d_beta_auto)
          + lapse_bh2 % msr % (ldnu%ldnu + ldbeta%ldbeta + 2.*ldldbeta)
          + lapse_bh2 % lapse_bh2 % msr % (2.*(1.+4.*msr) * ldbeta
                           - ldnu) / r_bh
          - (a_car % lapse_bh2 % lapse_bh2 % msr / nnn / r_bh)
          * (2.+3.*msr) * (3.+4.*msr) * lltkij
          + (2.*a_car % lapse_bh2 % lapse_bh2 % lapse_bh % msr
         / nnn / r_bh / r_bh) * (2.+10.*msr+9.*msr%msr) * lshift
          + (2.*pow(lapse_bh2, 3.) % msr % msr / r_bh / r_bh)
          % ((a_car%lapse_bh2/nnn/nnn - 1.) * pow(2.+3.*msr, 2.)
         + (a_car - 1.) * pow(1.+3.*msr, 2.)
         - 2.*(a_car%lapse_bh/nnn - 1.)*(2.+10.*msr+9.*msr%msr)) ;

        source.set_std_base() ;

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

        source().poisson_falloff(par_poisson2, beta_auto.set(),
                     k_falloff) ;

        // 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
        //----------------------------------------

        // Some auxiliary terms for the source
        // -----------------------------------

        Tenseur xx_con(mp, 1, CON, ref_triad) ;
        xx_con.set_etat_qcq() ;
        xx_con.set(0) = xx + sepa ;
        xx_con.set(1) = yy ;
        xx_con.set(2) = zz ;
        xx_con.set_std_base() ;

        Tenseur xsr_con(mp, 1, CON, ref_triad) ;
        xsr_con = xx_con / r_bh ;
        xsr_con.set_std_base() ;

        // Components of shift_auto with respect to the Cartesian triad
        //  (d/dx, d/dy, d/dz) of the mapping :

        Tenseur shift_auto_local = shift_auto ;
        shift_auto_local.change_triad( mp.get_bvect_cart() ) ;

        // Gradient (partial derivatives with respect to the Cartesian
        //           coordinates of the mapping)
        // dn(i, j) = D_i N^j

        Tenseur dn = shift_auto_local.gradient() ;

        // Return to the absolute reference frame
        dn.change_triad(ref_triad) ;

        // Trace of D_i N^j = divergence of N^j :
        Tenseur divn = contract(dn, 0, 1) ;

        // l^j D_j N^i
        Tenseur ldn_con = contract(xsr_con, 0, dn, 0) ;

        // D_j (l^k D_k N^i): dldn(j, i)
        Tenseur ldn_local = ldn_con ;
        ldn_local.change_triad( mp.get_bvect_cart() ) ;
        Tenseur dldn = ldn_local.gradient() ;
        dldn.change_triad(ref_triad) ;

        // l^j D_j (l^k D_k N^i)
        Tenseur ldldn = contract(xsr_con, 0, dldn, 0) ;

        // l_k D_j N^k
        Tenseur ldn_cov = contract(xsr_cov, 0, dn, 1) ;

        // l^j l_k D_j N^k
        Tenseur lldn_cov = contract(xsr_con, 0, ldn_cov, 0) ;

        // eta^{ij} l_k D_j N^k
        Tenseur eldn(mp, 1, CON, ref_triad) ;
        eldn.set_etat_qcq() ;
        eldn.set(0) = ldn_cov(0) ;
        eldn.set(1) = ldn_cov(1) ;
        eldn.set(2) = ldn_cov(2) ;
        eldn.set_std_base() ;

        // l^i D_j N^j
        Tenseur ldivn = xsr_con % divn ;

        // D_j (l^i D_k N^k): dldivn(j, i)
        Tenseur ldivn_local = ldivn ;
        ldivn_local.change_triad( mp.get_bvect_cart() ) ;
        Tenseur dldivn = ldivn_local.gradient() ;
        dldivn.change_triad(ref_triad) ;

        // l^j D_j (l^i D_k N^k)
        Tenseur ldldivn = contract(xsr_con, 0, dldivn, 0) ;

        // l_j N^j
        Tenseur ln = contract(xsr_cov, 0, shift_auto, 0) ;

        Tenseur vtmp =  6. * d_beta_auto - 8. * d_logn_auto ;

        Tenseur lvtmp = contract(xsr_con, 0, vtmp, 0) ;

        // eta^{ij} vtmp_j
        Tenseur evtmp(mp, 1, CON, ref_triad) ;
        evtmp.set_etat_qcq() ;
        evtmp.set(0) = vtmp(0) ;
        evtmp.set(1) = vtmp(1) ;
        evtmp.set(2) = vtmp(2) ;
        evtmp.set_std_base() ;

        // lapse_ns
        Tenseur lapse_ns(mp) ;
        lapse_ns = exp(logn_auto) ;
        lapse_ns.set_std_base() ;

        // Source
        // ------

        source_shift = (-4.*qpig) * nnn % a_car % (ener_euler + press)
          % u_euler
          + nnn % flat_scalar_prod_desal(tkij_auto, vtmp)
          - 2.*nnn % lapse_bh2 * msr / r_bh
          % flat_scalar_prod_desal(tkij_auto, xsr_cov)
          + 2.*lapse_bh2 * msr * (3.*ldldn + ldldivn) / 3.
          - lapse_bh2 * msr / r_bh
          * (4.*ldivn - lapse_bh2 % (3.*ldn_con + 8.*msr * ldn_con)
         - (eldn + 2.*lapse_bh2*(9.+11.*msr)*lldn_cov%xsr_con) / 3.)
          - 2.*lapse_bh2 % lapse_bh2 * msr / r_bh / r_bh
          * ( (4.+11.*msr) * shift_auto
          - lapse_bh2 * (12.+51.*msr+46.*msr*msr) * ln % xsr_con )
          / 3.
          + 8.*pow(lapse_bh2, 4.) * msr / r_bh / r_bh
          % (lapse_ns - 1.) * (2.+10.*msr+9.*msr*msr) * xsr_con / 3.
          + 2.*pow(lapse_bh2, 3.) * msr / r_bh * (2.+3.*msr)
          * ( (1.+2.*msr) * evtmp - (3.+2.*msr) * lvtmp * xsr_con) / 3. ;

        source_shift.set_std_base() ;

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

        // Filter for the source of shift vector :

        for (int i=0; i<3; i++) {
            for (int l=0; l<nz; l++) {
            if (source_shift(i).get_etat() != ETATZERO)
            source_shift.set(i).filtre_phi(np_filter, l) ;
        }
        }

        // 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) ;

        int* shift_falloff ;
        shift_falloff = new int[4] ;
        shift_falloff[0] = 1 ;
        shift_falloff[1] = 1 ;
        shift_falloff[2] = 2 ;
        shift_falloff[3] = 1 ;

        source_shift.poisson_vect_falloff(lambda_shift, par_poisson_vect,
                          shift_auto, w_shift,
                          khi_shift, shift_falloff) ;

        delete[] shift_falloff ;

        // Check: has the equation for shift_auto been correctly solved ?
        // --------------------------------------------------------------

        // Divergence of shift_auto :
        Tenseur divna = contract(shift_auto.gradient(), 0, 1) ;
        //      divna.dec2_dzpuis() ;    // dzpuis 2 -> 0

        // Grad(div) :
        Tenseur graddivn = divna.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_falloff 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                    //
    //========================================================//

}


void Et_bin_bhns_extr::equil_bhns_extr_cf(double ent_c, const double& mass,
                      const double& sepa, int mermax,
                      int mermax_poisson,
                      double relax_poisson,
                      int mermax_potvit,
                      double relax_potvit, int np_filter,
                      double thres_adapt,
                      Tbl& diff) {

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

    assert( kerrschild == false ) ;

    // 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 ;  // 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 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 ;

    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_pot_extr(mass, sepa, 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 ) ;

        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 ;

    //---------------------------------------------------------
    // Change the enthalpy field to accelerate the convergence
    //---------------------------------------------------------

    double dentdx = ent().dsdx()(0, 0, 0, 0) ;
    double dentdy = ent().dsdy()(0, 0, 0, 0) ;

    cout << "dH/dx|_center = " << dentdx << endl ;
    cout << "dH/dy|_center = " << dentdy << endl ;

    double dec_fact = 1. ;

    Tenseur func_in(mp) ;
    func_in.set_etat_qcq() ;
    func_in.set() = 1. - dec_fact * (dentdx/ent_c) * mp.x ;
    func_in.set().annule(nzet, nz-1) ;
    func_in.set_std_base() ;

    Tenseur func_ex(mp) ;
    func_ex.set_etat_qcq() ;
    func_ex.set() = 1. ;
    func_ex.set().annule(0, nzet-1) ;
    func_ex.set_std_base() ;

    // New enthalpy field
    // ------------------
    ent.set() = ent() * (func_in() + func_ex()) ;

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

    double dentdx_new = ent().dsdx()(0, 0, 0, 0) ;

    cout << "dH/dx|_new    = " << dentdx_new << endl ;

    //-----------------------------------------------------
    // Adaptation of the mapping to the new enthalpy field
    //----------------------------------------------------

    // Shall the adaptation be performed (cusp) ?
    // ------------------------------------------

    double dent_eq = ent().dsdr().val_point(ray_eq_pi(),M_PI/2.,M_PI) ;
    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 ;

    mp.adapt(ent(), par_adapt) ;

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

    mp_prev.homothetie(alpha_r) ;

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

    double fact_resize = 1. / alpha_r ;
    for (int l=nzet; l<nz-1; l++) {
        mp_et.resize(l, fact_resize) ;
    }
    mp_et.resize_extr(fact_resize) ;

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


    //----------------------------------------------------
    // Auxiliary terms for the source of Poisson equation
    //----------------------------------------------------

    const Coord& xx = mp.x ;
    const Coord& yy = mp.y ;
    const Coord& zz = mp.z ;

    Tenseur r_bh(mp) ;
    r_bh.set_etat_qcq() ;
    r_bh.set() = pow( (xx+sepa)*(xx+sepa) + yy*yy + zz*zz, 0.5) ;
    r_bh.set_std_base() ;

    Tenseur xx_cov(mp, 1, COV, ref_triad) ;
    xx_cov.set_etat_qcq() ;
    xx_cov.set(0) = xx + sepa ;
    xx_cov.set(1) = yy ;
    xx_cov.set(2) = zz ;
    xx_cov.set_std_base() ;

    Tenseur msr(mp) ;
    msr = ggrav * mass / r_bh ;
    msr.set_std_base() ;

    Tenseur tmp(mp) ;
    tmp = 1. / ( 1. - 0.25*msr*msr ) ;
    tmp.set_std_base() ;

    Tenseur xdnu(mp) ;
    xdnu = flat_scalar_prod_desal(xx_cov, d_logn_auto) ;
    xdnu.set_std_base() ;

    Tenseur xdbeta(mp) ;
    xdbeta = flat_scalar_prod_desal(xx_cov, d_beta_auto) ;
    xdbeta.set_std_base() ;

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

    // Source
    // ------

    if (relativistic) {

        source = qpig * a_car % (ener_euler + s_euler) + akcar_auto
          - flat_scalar_prod_desal(d_logn_auto, d_beta_auto)
          - 0.5 * tmp % msr % msr % xdnu / r_bh / r_bh
          - tmp % msr % xdbeta / r_bh / r_bh ;

    }
    else {
        cout << "The computation of BH-NS binary systems"
         << " should be relativistic !!!" << endl ;
        abort() ;
    }

    source.set_std_base() ;

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

    int k_falloff = 1 ;

    source().poisson_falloff(par_poisson1, logn_auto.set(), k_falloff) ;

    // Construct logn_auto_regu for et_bin_upmetr_extr.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
        // ------

        source = qpig * a_car % s_euler + 0.75 * akcar_auto
          - 0.5 * flat_scalar_prod_desal(d_logn_auto, d_logn_auto)
          - 0.5 * flat_scalar_prod_desal(d_beta_auto, d_beta_auto)
          - tmp % msr % xdnu / r_bh / r_bh
          - 0.5 * tmp % msr %msr % xdbeta / r_bh / r_bh ;

        source.set_std_base() ;

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

        source().poisson_falloff(par_poisson2, beta_auto.set(),
                     k_falloff) ;

        // 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
        //----------------------------------------

        // Some auxiliary terms for the source
        // -----------------------------------

        Tenseur bhtmp(mp, 1, COV, ref_triad) ;
        bhtmp.set_etat_qcq() ;
        bhtmp = tmp % msr % (3.*msr-8.) % xx_cov / r_bh / r_bh ;
        bhtmp.set_std_base() ;

        Tenseur vtmp =  6. * d_beta_auto - 8. * d_logn_auto ;

        // Source
        // ------

        source_shift = (-4.*qpig) * nnn % a_car % (ener_euler + press)
          % u_euler
          + nnn % flat_scalar_prod_desal(tkij_auto, vtmp+bhtmp) ;

        source_shift.set_std_base() ;

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

        // Filter for the source of shift vector :

        for (int i=0; i<3; i++) {
            for (int l=0; l<nz; l++) {
            if (source_shift(i).get_etat() != ETATZERO)
            source_shift.set(i).filtre_phi(np_filter, l) ;
        }
        }

        // 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) ;

        int* shift_falloff ;
        shift_falloff = new int[4] ;
        shift_falloff[0] = 1 ;
        shift_falloff[1] = 1 ;
        shift_falloff[2] = 2 ;
        shift_falloff[3] = 1 ;

        source_shift.poisson_vect_falloff(lambda_shift, par_poisson_vect,
                          shift_auto, w_shift,
                          khi_shift, shift_falloff) ;

        delete[] shift_falloff ;

        // Check: has the equation for shift_auto been correctly solved ?
        // --------------------------------------------------------------

        // Divergence of shift_auto :
        Tenseur divna = contract(shift_auto.gradient(), 0, 1) ;
        //      divna.dec2_dzpuis() ;    // dzpuis 2 -> 0

        // Grad(div) :
        Tenseur graddivn = divna.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_falloff 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                    //
    //========================================================//

}

---