Isol_hole/isol_hole_compute_metric.C

/*
 * Method of class Isol_Hole to compute metric data associated to a quasistationary single black hole spacetime slice.
 *
 * (see file isol_hole.h for documentation).
 *
 */

/*
 *   Copyright (c) 2009 Nicolas Vasset
 *
 *   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
 *
 */

// Headers C
#include "math.h"

// Headers Lorene
#include "tenseur.h"
#include "star.h"
#include "param.h"
#include "graphique.h"
#include "unites.h"     
#include "spheroid.h"
#include "excision_surf.h"
#include "excision_hor.h"
#include "param_elliptic.h"
#include "vector.h"
#include "scalar.h"
#include "diff.h"
#include "proto.h"
#include "isol_hole.h"


namespace Lorene {
void Isol_hole::compute_stat_metric(double precis, double relax, int mer_max, int mer_max2, bool isvoid){
    
    // Fundamental constants and units
    // -------------------------------
    using namespace Unites ;

    // Noticing the user that the iteration has started

    cout << "================================================================" << endl;
    cout << "STARTING THE MAIN ITERATION FOR COMPUTING METRIC FIELDS" << endl;
    cout << "        iteration parameters are the following:        " << endl;
    cout << "        convergence precision required:" << precis << endl;
    cout << "        max number of global steps    :" << mer_max << endl;
    cout << "        relaxation parameter          :" << relax   << endl;
    cout << "================================================================" << endl;


  // Construction of a multi-grid (Mg3d) and an affine mapping from the class mapping
  // --------------------------------------------------------------------------------
  const Map_af* map = dynamic_cast<const Map_af*>(&mp) ;
  const Mg3d* mgrid = (*map).get_mg();
    
  // Construct angular grid for h(theta,phi) 
  const Mg3d* g_angu = (*mgrid).get_angu_1dom() ;
  
  const int nz = (*mgrid).get_nzone();  // Number of domains
  int nt = (*mgrid).get_nt(1);  // Number of collocation points in theta in each domain
  const int np = (*mgrid).get_np(1); 
  const Coord& rr = (*map).r;
   Scalar rrr (*map) ; 
  rrr = rr ; 
  rrr.std_spectral_base();  
  assert((rrr.val_grid_point(1,0,0,0) - 1.) <= 1.e-9); // For now the code handles only horizons at r=1, corresponding to the first shell inner boundary. This test assures this is the case with our mapping.
 
  // Angular mapping defined as well
  //--------------------------------
  double r_limits2[] = {rrr.val_grid_point(1,0,0,0), rrr.val_grid_point(2,0,0,0)} ; 
  const Map_af map_2(*g_angu, r_limits2); //2D mapping; check if this is useful.
  const Metric_flat& mets = (*map).flat_met_spher() ;
 
    //----------------
    // Initializations
    // ---------------



  Scalar logn (*map) ; logn = log(lapse) ; 
  logn.std_spectral_base();


  Scalar logpsi(*map) ; logpsi = log(conf_fact) ;
  logpsi.std_spectral_base();
  Scalar psi4 (*map) ; psi4 = conf_fact*conf_fact*conf_fact*conf_fact ;
  Scalar npsi (*map) ;  npsi =conf_fact*lapse ;
  

  Scalar conf_fact_new(*map) ; conf_fact_new.annule_hard(); conf_fact_new.std_spectral_base(); 
  Scalar npsi_new(*map); npsi_new.annule_hard(); npsi_new.std_spectral_base();

  Vector shift_new (*map, CON, (*map).get_bvect_spher()); 
  for(int i=1; i<=3; i++){
    shift_new.set(i)=0;
  }
  shift_new.std_spectral_base();
 
  // Non conformally flat variables

 
  Sym_tensor gamtuu = mets.con() + hij; 
  Metric gamt(gamtuu);
  Metric gam(gamt.cov()*psi4) ;
  Sym_tensor gamma = gam.cov();
  

  // Extrinsic curvature variables

  Sym_tensor aa(*map, CON, (*map).get_bvect_spher());
  for (int iii= 1; iii<=3; iii++){ 
    for(int j=1; j<=3; j++){
      aa.set(iii,j)= 0;
    }
  }
  aa.std_spectral_base(); 
  Scalar aa_quad_scal(*map) ; aa_quad_scal = 0. ;

  Sym_tensor aa_hat(*map, CON, (*map).get_bvect_spher());
  for (int iii= 1; iii<=3; iii++){ 
    for(int j=1; j<=3; j++){
      aa_hat.set(iii,j)= 0;
    }
  }
 
  Sym_tensor kuu = aa/psi4 ;
  Sym_tensor kuu2 = aa_hat/(psi4*psi4*sqrt(psi4));
  Sym_tensor kdd = contract (gamma, 0, contract(gamma, 1, kuu, 0),1);
 

  // (2,1)-rank delta tensor: difference between ricci rotation coefficients. 

   Tensor delta = -0.5*contract( hij, 1, gamt.cov().derive_cov(mets), 2);
   Scalar tmp(*map);

   for (int i=1; i<=3; i++) {
     for (int j=1; j<=3; j++) {
       for (int k=1; k<=3; k++) {
     tmp = 0.;
     tmp =  -0.5 *(gamt.cov().derive_con(mets))(i,j,k);
     for (int l=1; l<=3; l++) {
       tmp += -0.5*( gamt.cov()(i,l)*(hij.derive_cov(mets))(k,l,j) + gamt.cov()(l,j)*(hij.derive_cov(mets))(k,l,i));
     }
         delta.set(k,i,j) += tmp ; 
       }
     }
   }
   

   // Conformal Rstar scalar(eq 61, Bonazzola et al. 2003) 
   Scalar Rstar = 
    0.25 * contract(gamt.con(), 0, 1,
            contract(hij.derive_cov(mets), 0, 1, gamt.cov().derive_cov(mets), 0, 1), 0, 1 ) 
    - 0.5  * contract(gamt.con(), 0, 1,
              contract(hij.derive_cov(mets), 0, 1, gamt.cov().derive_cov(mets), 0, 2), 0, 1 ) ; 
  
   Scalar norm(*map);
   Scalar norm3(*map);

   
   if (isvoid == false){
   
     cout <<"FAIL: case of non-void spacetime not treated yet" << endl;
   }
 
   else
     {

  // Parameters for the iteration 


  double diff_ent = 1 ; // initialization of difference marker between two iterations; 

  int util = 0; // Tool used to stop tensorial iteration at any wished step "util"
  
  
  
  for(int mer=0 ;(diff_ent > precis) && (mer<mer_max) ; mer++) {    

   //Global relaxation coefficient

    // Scalar variables linked to the norm of normal vector to horizon.
    norm = sqrt(1. + hij(1,1)); norm.std_spectral_base();
    norm3 = sqrt(1. + hij(3,3)); norm3.std_spectral_base();
    





      // Solving for (Psi-1) //

  
      // Setting of the boundary

      double   diric= 0.; 
      double   neum = 1.; 

      Vector ssalt = rrr.derive_cov(gam);
      Vector ssaltcon = ssalt.up_down(gam);
      Scalar ssnormalt = sqrt(contract (ssalt,0, ssaltcon, 0));
      ssnormalt.std_spectral_base();
      
      ssalt.annule_domain(nz-1);
      ssalt.annule_domain(0);
      ssaltcon.annule_domain(nz-1);
      ssaltcon.annule_domain(0);
      
      ssalt = ssalt/ssnormalt;
      ssaltcon = ssaltcon/ssnormalt;
      Vector ssconalt = ssaltcon*conf_fact*conf_fact;    // \tilde{s} in the notations of Gourgoulhon and Jaramillo, 2006
      ssconalt.std_spectral_base();
      ssconalt.annule_domain(nz-1);
      Scalar bound3bis =   -((1./conf_fact)*contract((contract(kdd,1,ssconalt,0)),0, ssconalt,0));
  
      bound3bis.annule_domain(nz-1);
      bound3bis += -conf_fact*ssconalt.divergence(gamt);
      bound3bis.annule_domain(nz-1);     
      bound3bis = 0.25*bound3bis;
      bound3bis += -contract(conf_fact.derive_cov(gamt),0,ssconalt,0) + conf_fact.dsdr();
      bound3bis.annule_domain(nz-1);
      bound3bis.std_spectral_base();
      bound3bis.set_spectral_va().ylm();    

      Mtbl_cf *boundd3bis = bound3bis.set_spectral_va().c_cf;
            




      // Computing the source
   
      Scalar source_conf_fact(*map) ; source_conf_fact=3. ; // Pour le fun... 
      source_conf_fact.std_spectral_base();                            

      Scalar d2logpsi = contract(conf_fact.derive_cov(mets).derive_cov(mets), 0, 1, hij, 0,1);  
      d2logpsi.inc_dzpuis(1);
 
      source_conf_fact = -(0.125* aa_quad_scal )/(psi4*conf_fact*conf_fact*conf_fact) +  conf_fact* 0.125* Rstar - d2logpsi; 
      
      source_conf_fact.std_spectral_base(); 

      if (source_conf_fact.get_etat() == ETATZERO) {
    source_conf_fact.annule_hard() ;
    source_conf_fact.set_dzpuis(4) ;
    source_conf_fact.std_spectral_base() ;
      }
      source_conf_fact.set_spectral_va().ylm();
    
      


      // System inversion   

      Param_elliptic source11(source_conf_fact);
      conf_fact_new = source_conf_fact.sol_elliptic_boundary(source11, *boundd3bis, diric , neum) + 1 ; // Resolution has been done for quantity Q-1, because our solver gives a vanishing solution at infinity! 
      

 
    // tests for resolution

    Scalar baba2 = (conf_fact_new-1).laplacian();
//     cout << "psi+1-résolution" << endl;
//     maxabs (baba2 - source_conf_fact);
   
    Scalar psinewbis = conf_fact_new -1. ; psinewbis.annule_domain(nz -1);
    psinewbis.std_spectral_base();
    psinewbis = psinewbis.dsdr();
    Scalar psinewfin2 (map_2) ;
    psinewfin2.allocate_all(); 
    psinewfin2.set_etat_qcq();  
    psinewfin2.std_spectral_base();
    
    for (int k=0; k<np; k++)
      for (int j=0; j<nt; j++) {
    
    psinewfin2.set_grid_point(0, k, j, 0) = psinewbis.val_grid_point(1, k,j,0) - bound3bis.val_grid_point(1, k, j, 0);
    
      }
    
    //  maxabs (psinewfin2);
    


    // Update during the loop

    conf_fact = conf_fact_new* (1-relax) + conf_fact* relax ;
    psi4 = conf_fact*conf_fact*conf_fact*conf_fact;
    logpsi = log(conf_fact) ; 
    logpsi.std_spectral_base();    







    // Solving for (N*Psi -1)/ 


    // Setting of the boundary
    assert (NorKappa == false) ; 
    Scalar bound(*map);
    bound = (boundNoK)*conf_fact -1;
    bound.annule_domain(nz -1);
    bound.std_spectral_base();
    bound.set_spectral_va().ylm();
    Mtbl_cf *boundd = bound.get_spectral_va().c_cf;

     diric =1; 
     neum = 0 ; 


    

    // Computing the source ...      
         Scalar d2lognpsi = contract(npsi.derive_cov(mets).derive_cov(mets), 0, 1, hij, 0,1);
       d2lognpsi.inc_dzpuis(1); //  dzpuis correction.
  
       Scalar source_npsi = npsi*(aa_quad_scal*(7./8.)/(psi4*psi4) + Rstar/8.) - d2lognpsi; // ?
    source_npsi.std_spectral_base();
    if (source_npsi.get_etat() == ETATZERO) {
      source_npsi.annule_hard() ;
      source_npsi.set_dzpuis(4) ;
      source_npsi.std_spectral_base() ;
    }



    // Inversion of the operator
    Param_elliptic source1 (source_npsi); 
    npsi_new = source_npsi.sol_elliptic_boundary(source1, *boundd, diric, neum) ;

    npsi_new = npsi_new +1;

  

    // Resolution tests in npsi
    Scalar baba = npsi_new.laplacian();
//       cout << "resolution_npsi" << endl;
//      maxabs (baba - source_npsi);


    //  cout << "bound_npsi" << endl;
        Scalar npsibound2 (map_2) ;
    npsibound2.allocate_all(); 
    npsibound2.set_etat_qcq();  
    npsibound2.std_spectral_base();      
    for (int k=0; k<np; k++)
      for (int j=0; j<nt; j++) {
             
    npsibound2.set_grid_point(0, k, j, 0) = npsi_new.val_grid_point(1, k,j,0) - bound.val_grid_point(1, k, j, 0) -1.;
             
      }

    //   maxabs (npsibound2);
    


    // Update during the loop

      
      npsi = npsi_new*(1-relax) + npsi* relax; 
       lapse = npsi/conf_fact; 
       logn = log(lapse);
       logn.std_spectral_base(); 

 

      //Resolution in Beta //


      // Setting of the boundary  
           
      bound = (boundNoK)/(conf_fact*conf_fact) ;
      bound.annule_domain(nz -1);
 

      Scalar hor_rot(*map); hor_rot.annule_hard(); hor_rot = Omega; // Rotation parameter for spacetime
      hor_rot.std_spectral_base() ; hor_rot.mult_rsint();
      hor_rot.annule_domain(nz -1);

      Vector limit = shift_new;
      Vector ephi(*map, CON, (*map).get_bvect_spher());
      ephi.set(1).annule_hard();
      ephi.set(2).annule_hard();
      ephi.set(3) = 1;
      ephi.std_spectral_base();
      ephi.annule_domain(nz -1);

      limit = bound*ssconalt + hor_rot*ephi;
      limit.std_spectral_base(); // Boundary is fixed by value of 3 components of a vector (rather than value of potentials)   

      Scalar Vrb = limit(1);  Vrb.set_spectral_va().ylm();
      Scalar mmub = limit.mu(); mmub.set_spectral_va().ylm(); 
      Scalar etab = limit.eta(); etab.set_spectral_va().ylm();



 
     // Computing the source
       Vector deltaA =  - 2*lapse*contract(delta, 1,2, aa, 0,1);
       Vector hijddb =  - contract (shift.derive_cov(mets).derive_cov(mets), 1,2, hij, 0,1) ;
       Vector hijddivb =  - 0.3333333333333* contract (shift.divergence(mets).derive_cov(mets),0, hij,1);
       hijddb.inc_dzpuis(); // dzpuis fixing patch... 
       hijddivb.inc_dzpuis(); 
       
       Vector sourcevect2(*map,CON, (*map).get_bvect_spher()); 
      sourcevect2= (2.* contract(aa, 1, lapse.derive_cov(mets),0)-12*lapse*contract(aa, 1, logpsi.derive_cov(mets), 0))  + deltaA + hijddb + hijddivb ; 
       
      sourcevect2.set(1).set_dzpuis(4);
      sourcevect2.set(2).set_dzpuis(4);
      sourcevect2.set(3).set_dzpuis(4);
      sourcevect2.std_spectral_base(); 
      if(sourcevect2.eta().get_etat() == ETATZERO)
    { sourcevect2.set(2).annule_hard();}
 
      
      double lam = (1./3.);    
    
   


      // System inversion
      sourcevect2.poisson_boundary2(lam, shift_new, Vrb, etab, mmub, 1., 0., 1. ,0. ,1. ,0.) ;   



    
    // resolution tests
    Vector source2 = contract(shift_new.derive_con(mets).derive_cov(mets), 1,2) + lam* contract(shift_new.derive_cov(mets), 0,1).derive_con(mets);
    source2.inc_dzpuis(1);
    //  maxabs (source2 - sourcevect2);   

    Scalar mufin = shift_new.mu();
    mufin.set_spectral_va().coef();
         
    Scalar mufin2 (map_2) ;
    mufin2.allocate_all(); 
    mufin2.set_etat_qcq();  
    mufin2.std_spectral_base();
         
    for (int k=0; k<np; k++)
      for (int j=0; j<nt; j++) {
             
    mufin2.set_grid_point(0, k, j, 0) = mufin.val_grid_point(1, k,j,0) - mmub.val_grid_point(1, k, j, 0);
             
      }

    //  maxabs (mufin2);


    Scalar brfin = shift_new(1);
    brfin.set_spectral_va().coef();
         
    Scalar brfin2 (map_2) ;
    brfin2.allocate_all(); 
    brfin2.set_etat_qcq();  
    brfin2.std_spectral_base();
         
    for (int k=0; k<np; k++)
      for (int j=0; j<nt; j++) {
             
    brfin2.set_grid_point(0, k, j, 0) = brfin.val_grid_point(1, k,j,0) - Vrb.val_grid_point(1, k, j, 0);
             
      }
    //  maxabs (brfin2);




    // Update during the loop
    for (int ii=1; ii <=3; ii++){
     shift.set(ii) = shift_new(ii)*(1-relax) + shift(ii)* relax;
    }
       

    diff_ent = max(maxabs(npsi_new - npsi )); // Convergence parameter (discutable relevance...) 





  
    // Tensor hij resolution        ///

    if (isCF == false){
   
    if (diff_ent <=5.e-3) { // No resolution until we are close to the result.
   
   //WARNING; parameter maybe to be changed according to convergence parameters in Poisson-Hole/Kerr1.9/pplncp.C
      util = util+1; // Loop marker for NCF equation.
     


     // Local convergence can be asked for hij equation. Allows to satisfy integrability conditions.
  double precis2 =  1.e5*precis ; // Here we ask for a local convergence; this can be improved later. WARNING; parameter maybe to be changed according to convergence parameters in Poisson-Hole/Kerr1.9/pplncp.C

  double diff_ent2 = 1 ; // Local convergence marker
  double relax2 = 0.5; // Local relaxation parameter. If not 1, the determinant condition won't be satisfied on a particular iteration.


  Sym_tensor sourcehij = hij; // Random initialization...

  for(int mer2=0 ;(diff_ent2 > precis2) && (mer2<mer_max2) ; mer2++) {    


    // Calculation of the source

    // The double Lie derivative term is taken care of in the subroutine. 

       secmembre_kerr(sourcehij);
       
      
    //System inversion (note that no boundary condition is imposed)


     Sym_tensor hij_new = hij;

     hij_new = boundfree_tensBC (sourcehij, shift , conf_fact, lapse, hij, precis2);

     cout << "maximum of convergence marker for the subiteration" << endl;

     diff_ent2 = max(maxabs(hij - hij_new));
      hij = relax2*hij_new + (1 - relax2)*hij;

     cout << "mer2, diffent2" << endl;
 
      cout << mer2 << endl;
      cout << diff_ent2 << endl;


  }

   // Resolution tests


       Sym_tensor gammatilde = mets.con() + hij;   
       Metric gammatilde2(gammatilde); Scalar detgam = gammatilde2.determinant();
 //       cout << "determinant of result" << endl;
//       maxabs (detgam-1.);
   
//      cout << "comment l'equation en hij est elle vérifiée?" << endl;
        
      Sym_tensor test =contract (hij.derive_cov(mets).derive_con(mets), 2,3);
      test.annule(nz-1, nz-1);
      test = test - hij.derive_lie(shift).derive_lie(shift)/ ((lapse/(conf_fact*conf_fact))*(lapse/(conf_fact*conf_fact)));        
      test.annule(nz-1, nz-1);
      Sym_tensor youps = test - sourcehij/((lapse/(conf_fact*conf_fact))*(lapse/(conf_fact*conf_fact)));
//       maxabs (youps); 
//       maxabs((youps).trace(mets));
//       cout << " AAABBB" << endl;
//       maxabs((youps).compute_A());
//       maxabs((youps).compute_tilde_B());

    
    }
    }
 




    

     // Global variable update after an entire loop ////

   
       gamtuu = mets.con() + hij;
       gamt = gamtuu; // Métrique.
       gam = gamt.cov()*psi4;
       gamma = gam.cov();
       

       for (int i=1; i<=3; i++) {
     for (int j=1; j<=3; j++) {
       
       tmp = 0;       
       tmp = ((shift.derive_con(mets))(i,j) + (shift.derive_con(mets))(j,i) - (2./3.)*shift.divergence(mets) * mets.con()(i,j))*(1./(2.*lapse));       
       aa.set(i,j) = tmp ; 
       
     }
       }

       aa = aa -  (hij.derive_lie(shift) + (2./3.)*shift.divergence(mets)*hij)*(1./(2.*lapse)); //Non conformally flat correction; we assume here dhij/dt = 0.
  
       aa_hat = aa*psi4*sqrt(psi4); // Rescaling of traceless exrinsic curvature.
       aa_hat.std_spectral_base();


       hatA = aa_hat; // Probably obsolete very soon... replace aa_hat by hatA.

       Sym_tensor aaud = aa.up_down(gamt);
       Sym_tensor aaud_hat = aa_hat.up_down(gamt);
       aa_quad_scal =  contract(contract (aa_hat, 0, aaud_hat, 0), 0,1);

       delta = -0.5*contract( hij, 1, gamt.cov().derive_cov(mets), 2);
 

   for (int i=1; i<=3; i++) {
     for (int j=1; j<=3; j++) {
       for (int k=1; k<=3; k++) {
     tmp = 0.;
     tmp =  -0.5 *(gamt.cov().derive_con(mets))(i,j,k);
     for (int l=1; l<=3; l++) {
       tmp += -0.5*( gamt.cov()(i,l)*(hij.derive_cov(mets))(k,l,j) + gamt.cov()(l,j)*(hij.derive_cov(mets))(k,l,i));
     }
         delta.set(k,i,j) += tmp ; 
       }
     }
   } 
   
  Rstar = 
    0.25 * contract(gamt.con(), 0, 1,
            contract(gamt.con().derive_cov(mets), 0, 1, gamt.cov().derive_cov(mets), 0, 1), 0, 1 ) 
    - 0.5  * contract(gamt.con(), 0, 1,
              contract(gamt.con().derive_cov(mets), 0, 1, gamt.cov().derive_cov(mets), 0, 2), 0, 1 ) ;  

    kuu = aa/(psi4); 
    kdd =  contract (gamma, 0, contract(gamma, 1, kuu, 0),1);  
 



  // Convergence markers
    cout << "diffent" << endl; 
    cout<< diff_ent << endl;   
    cout <<"mer" << mer << endl;   
   



     
  //------------------------------------------------
  //     Check of Einstein equations (3+1 form)
  //------------------------------------------------


  // Lapse 
  //------
  Scalar lapse2(*map) ;
  lapse2 = lapse ;
  lapse2.std_spectral_base();

  // 3-metric
  //---------
  //   const Metric gam(mets.cov()*psi4) ;
   
  Sym_tensor gamt2(*map, COV, (*map).get_bvect_spher()); 
  for (int i=1; i<=3; i++)
    for (int j=1; j<=3; j++) 
      { gamt2.set(i,j)=gam.cov()(i,j); 
      }
  //Shift
  //-----
  Vector beta = shift ;
  
  // Extrinsic curvature
  //--------------------

  Scalar TrK3(*map);
  Sym_tensor k_uu = aa/(psi4) ;
  k_uu.dec_dzpuis(k_uu(1,1).get_dzpuis()); 
  Sym_tensor k_dd = k_uu.up_down(gam); // Another way of computing the same thing, just to be sure... 

  TrK3 = k_uu.trace(gam);
  //  TrK3.spectral_display("TraceKvraie", 1.e-10);
 
  // Hamiltonian constraint
  //-----------------------
  Scalar ham_constr = gam.ricci_scal() ;
  ham_constr.dec_dzpuis(3) ;
  ham_constr +=  TrK3*TrK3 - contract(k_uu, 0, 1, k_dd, 0, 1) ;
  // maxabs(ham_constr, "Hamiltonian constraint: ") ;

  ham_constr.set_spectral_va().ylm();
  //  ham_constr.spectral_display("ham_constr", 1.e-9);
  

 
  // Momentum constraint
  //-------------------
  Vector mom_constr = k_uu.divergence(gam)  - TrK3.derive_con(gam) ;
  mom_constr.dec_dzpuis(2) ;
//   maxabs(mom_constr, "Momentum constraint: ") ;
//   mom_constr(1).spectral_display("mom1", 1.e-9) ;
//  mom_constr(2).spectral_display("mom2", 1.e-9) ;
//  mom_constr(3).spectral_display("mom3", 1.e-9) ;
  
 
  // Evolution equations
  //--------------------
  Sym_tensor evol_eq = lapse2*gam.ricci() 
    - lapse2.derive_cov(gam).derive_cov(gam);
  evol_eq.dec_dzpuis() ;
  evol_eq += k_dd.derive_lie(beta) ;
  evol_eq.dec_dzpuis(2) ;
  evol_eq += lapse2*(TrK3*k_dd - 2*contract(k_dd, 1, k_dd.up(0, gam), 0) ) ;
//   maxabs(evol_eq, "Evolution equations: ") ;
  
//   evol_eq.trace(gam).spectral_display("evoltrace", 1.e-10);
//   maxabs (evol_eq.trace(gam));
 
 
  //  evol_eq.spectral_display("evol", 1.e-10);
  

  }

     

    cout << "================================================================" << endl;
    cout << "                THE ITERATION HAS NOW CONVERGED" << endl;
    cout << "================================================================" << endl;
     }
  return; 
}











}