LORENE
tslice_dirac_max_evolve.C
1 /*
2  * Method of class Tslice_dirac_max for time evolution
3  *
4  * (see file time_slice.h for documentation).
5  *
6  */
7 
8 /*
9  * Copyright (c) 2004 Eric Gourgoulhon & Jerome Novak
10  *
11  * This file is part of LORENE.
12  *
13  * LORENE is free software; you can redistribute it and/or modify
14  * it under the terms of the GNU General Public License version 2
15  * as published by the Free Software Foundation.
16  *
17  * LORENE is distributed in the hope that it will be useful,
18  * but WITHOUT ANY WARRANTY; without even the implied warranty of
19  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20  * GNU General Public License for more details.
21  *
22  * You should have received a copy of the GNU General Public License
23  * along with LORENE; if not, write to the Free Software
24  * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
25  *
26  */
27 
28 char tslice_dirac_max_evolve_C[] = "$Header: /cvsroot/Lorene/C++/Source/Time_slice/tslice_dirac_max_evolve.C,v 1.22 2015/08/10 15:32:27 j_novak Exp $" ;
29 
30 /*
31  * $Id: tslice_dirac_max_evolve.C,v 1.22 2015/08/10 15:32:27 j_novak Exp $
32  * $Log: tslice_dirac_max_evolve.C,v $
33  * Revision 1.22 2015/08/10 15:32:27 j_novak
34  * Better calls to Param::add_int(), to avoid weird problems (e.g. with g++ 4.8).
35  *
36  * Revision 1.21 2014/10/13 08:53:48 j_novak
37  * Lorene classes and functions now belong to the namespace Lorene.
38  *
39  * Revision 1.20 2013/01/24 12:55:18 j_novak
40  * Corrected the declaration of variables for boundary conditions.
41  *
42  * Revision 1.19 2012/02/06 12:59:07 j_novak
43  * Correction of some errors.
44  *
45  * Revision 1.18 2011/07/22 13:21:02 j_novak
46  * Corrected an error on BC treatment.
47  *
48  * Revision 1.17 2010/10/20 07:58:10 j_novak
49  * Better implementation of the explicit time-integration. Not fully-tested yet.
50  *
51  * Revision 1.16 2008/12/04 18:22:49 j_novak
52  * Enhancement of the dzpuis treatment + various bug fixes.
53  *
54  * Revision 1.15 2008/12/02 15:02:22 j_novak
55  * Implementation of the new constrained formalism, following Cordero et al. 2009
56  * paper. The evolution eqs. are solved as a first-order system. Not tested yet!
57  *
58  * Revision 1.13 2004/06/14 20:51:37 e_gourgoulhon
59  * Method solve_hij has now argument method_poisson.
60  * Its value is set **provisory** to 1 (instead of method_poisson_vect !).
61  *
62  * Revision 1.12 2004/05/31 09:09:59 e_gourgoulhon
63  * Added monitoring of khi and mu.
64  * Added writing of whole configuration in file (via Time_slice::save).
65  *
66  * Revision 1.11 2004/05/24 20:58:05 e_gourgoulhon
67  * Added graphical output of khi, mu and trh.
68  *
69  * Revision 1.10 2004/05/20 20:32:01 e_gourgoulhon
70  * Added arguments check_mod and save_mod.
71  * Argument graph_device passed to des_evol.
72  *
73  * Revision 1.9 2004/05/17 19:55:10 e_gourgoulhon
74  * Added arguments method_poisson_vect, nopause and graph_device
75  *
76  * Revision 1.8 2004/05/13 21:35:30 e_gourgoulhon
77  * Added monitoring of various quantities (as Evolution_full<Tbl>).
78  * Added function monitor_scalar.
79  *
80  * Revision 1.7 2004/05/12 15:24:20 e_gourgoulhon
81  * Reorganized the #include 's, taking into account that
82  * time_slice.h contains now an #include "metric.h".
83  *
84  * Revision 1.6 2004/05/11 20:15:10 e_gourgoulhon
85  * Added Evolution_full's for ADM mass and checks of the constraint,
86  * as well as the corresponding plots and write to files.
87  *
88  * Revision 1.5 2004/05/10 09:19:27 e_gourgoulhon
89  * Added a call to del_deriv() after set_khi_mu.
90  *
91  * Revision 1.4 2004/05/09 20:59:06 e_gourgoulhon
92  * Change of the time scheme: first solve d'Alembert equations,
93  * then psuh forward in time and solve the elliptic equation
94  * on the new slice.
95  *
96  * Revision 1.3 2004/05/06 15:26:29 e_gourgoulhon
97  * No longer necessary to initialize khi and mu.
98  *
99  * Revision 1.2 2004/05/05 14:39:32 e_gourgoulhon
100  * Added graphical outputs.
101  *
102  * Revision 1.1 2004/05/03 14:49:10 e_gourgoulhon
103  * First version
104  *
105  *
106  * $Header: /cvsroot/Lorene/C++/Source/Time_slice/tslice_dirac_max_evolve.C,v 1.22 2015/08/10 15:32:27 j_novak Exp $
107  *
108  */
109 
110 // Lorene headers
111 #include "time_slice.h"
112 #include "param.h"
113 #include "graphique.h"
114 #include "utilitaires.h"
115 #include "proto.h"
116 
117 namespace Lorene {
118 const Tbl& monitor_scalar(const Scalar& uu, Tbl& resu) ;
119 
120 void Tslice_dirac_max::evolve(double pdt, int nb_time_steps,
121  int niter_elliptic, double relax,
122  int check_mod, int save_mod,
123  int method_poisson_vect, int nopause,
124  const char* graph_device, bool verbose,
125  const Scalar* ener_euler,
126  const Vector* mom_euler, const Scalar* s_euler,
127  const Sym_tensor* strain_euler) {
128 
129  // Intermediate quantities
130  // -----------------------
131  const Map& map = nn().get_mp() ;
132  const Base_vect& triad = *(beta().get_triad()) ;
133  assert( triad == map.get_bvect_spher() ) ;
134 
135  // For graphical outputs:
136  int ngraph0 = 20 ; // index of the first graphic device to be used
137  int ngraph0_mon = 70 ; // for monitoring global quantities
138  int nz = map.get_mg()->get_nzone() ;
139  int nz_bound = nz - 2 ;
140  int nt = map.get_mg()->get_nt(0) ;
141  int np = map.get_mg()->get_np(0) ;
142  int np2 = np+2 ;
143  Scalar tmp(map) ; tmp.std_spectral_base() ;
144  Base_val base_ref = tmp.get_spectral_base() ;
145  Base_val base_pseudo = base_ref ;
146  base_pseudo.mult_cost() ;
147  base_pseudo.mult_x() ;
148 #ifndef NDEBUG
149  for (int lz=1; lz<nz; lz++) {
150  assert( map.get_mg()->get_np(lz) == np ) ;
151  assert( map.get_mg()->get_nt(lz) == nt ) ;
152  }
153  assert (depth > 2) ; //## for the moment, a more flexible test should be put
154  for (int it=0; it<depth; it++) {
155  assert ( hh_evol.is_known( jtime - it ) ) ;
156  assert ( hata_evol.is_known( jtime - it ) ) ;
157  assert ( A_hata_evol.is_known( jtime - it ) ) ;
158  assert ( B_hata_evol.is_known( jtime - it ) ) ;
159  assert ( A_hh_evol.is_known( jtime - it ) ) ;
160  assert ( B_hh_evol.is_known( jtime - it ) ) ;
161  }
162 #endif
163 
164  // Initialization of the TT fields
165  //--------------------------------
166  Sym_tensor_tt hij_tt( map, triad, ff ) ;
168  Sym_tensor_tt hijtt_old = hij_tt ;
169  for (int i=1; i<=3; i++)
170  for (int j=i; j<=3; j++)
171  if ( hijtt_old(i,j).get_etat() == ETATZERO )
172  hijtt_old.set( i, j ).annule_hard() ;
173  hijtt_old.annule(0, nz_bound) ;
174 
175  Sym_tensor_tt hata_tt( map, triad, ff ) ;
177  hata_tt.inc_dzpuis(2) ;
178  Sym_tensor_tt hatatt_old = hata_tt ;
179  for (int i=1; i<=3; i++)
180  for (int j=i; j<=3; j++)
181  if ( hatatt_old(i,j).get_etat() == ETATZERO )
182  hatatt_old.set( i, j ).annule_hard() ;
183  hatatt_old.annule(0, nz_bound) ;
184 
185  // Declaration / initialization of mu and khi for hh and hata
186  //-----------------------------------------------------------
187  Evolution_std<Scalar> khi_hh_evol(depth) ;
188  Evolution_std<Scalar> mu_hh_evol(depth) ;
189  Evolution_std<Scalar> khi_a_evol(depth) ;
190  Evolution_std<Scalar> mu_a_evol(depth) ;
191  Sym_tensor_trans Tij(map, map.get_bvect_spher(), ff) ;
192  for (int j=jtime-depth+1; j<=jtime; j++) {
193  tmp = hij_tt(1,1) ;
194  tmp.mult_r() ; tmp.mult_r() ;
195  khi_hh_evol.update( tmp, j, the_time[j] ) ;
196  mu_hh_evol.update( hij_tt.mu(), j, the_time[j] ) ;
197  tmp = hata_tt(1,1) ;
198  tmp.mult_r() ; tmp.mult_r() ;
199  khi_a_evol.update( tmp, j, the_time[j] ) ;
200  mu_a_evol.update( hata_tt.mu(), j, the_time[j] ) ;
201  }
202 
203  double Rmax = map.val_r(nz-2, 1., 0., 0.) ; // outermost radius
204  double ray_des = 1.25 * Rmax ; // for plots
205 
206  // Parameters for the evolution equations
207  //---------------------------------------
208  double an = 23./12. ;
209  double anm1 = -4./3. ;
210  double anm2 = 5./12. ;
211 
212  int i_zero = 0 ;
213  int i_minus_one = -1 ;
214  int i_two = 2 ;
215 
216  Param par_A ;
217  double *l_val_A = new double(1./Rmax) ;
218  double *l_der_A = new double(1.) ;
219  par_A.add_int(nz_bound, 0) ;
220  par_A.add_int(i_two, 1) ; //matching of function and derivative
221  par_A.add_int(i_zero, 2) ;// no shift in l
222  par_A.add_int(i_two, 3) ; // only for l>=2
223  par_A.add_double_mod(*l_val_A, 0) ;
224  par_A.add_double_mod(*l_der_A, 1) ;
225  Tbl* tmp_Acc = new Tbl(np2, nt) ;
226  Tbl& Acc = *tmp_Acc ;
227  Acc.annule_hard() ;
228  par_A.add_tbl_mod(Acc) ;
229  Param par_mat_A_hh ;
230 
231  Param par_B ;
232  double* l_val_B = new double(1./Rmax) ;
233  double* l_der_B = new double(1.) ;
234  par_B.add_int(nz_bound, 0) ;
235  par_B.add_int(i_two, 1) ; //matching of function and derivative
236  par_B.add_int(i_minus_one, 2) ;// shift in l for tilde{B}
237  par_B.add_int(i_two, 3) ; // only for l>=2
238  par_B.add_double_mod(*l_val_B, 0) ;
239  par_B.add_double_mod(*l_der_B, 1) ;
240  Tbl* tmp_Bcc = new Tbl(np2, nt) ;
241  Tbl& Bcc = *tmp_Bcc ;
242  Bcc.annule_hard() ;
243  par_B.add_tbl_mod(Bcc) ;
244  Param par_mat_B_hh ;
245 
246  Tbl xij_b(np2, nt) ;
247  xij_b.set_etat_qcq() ;
248  initialize_outgoing_BC(nz_bound, B_hh_evol[jtime] , B_hata_evol[jtime], xij_b) ;
249  Tbl xijm1_b(np2, nt) ;
250  xijm1_b.set_etat_qcq() ;
251  initialize_outgoing_BC(nz_bound, B_hh_evol[jtime-1] ,
252  B_hata_evol[jtime-1], xijm1_b) ;
253  Tbl xij_a(np2, nt) ;
254  xij_a.set_etat_qcq() ;
255  initialize_outgoing_BC(nz_bound, A_hh_evol[jtime] , A_hata_evol[jtime], xij_a) ;
256  Tbl xijm1_a(np2, nt) ;
257  xijm1_a.set_etat_qcq() ;
258  initialize_outgoing_BC(nz_bound, A_hh_evol[jtime-1] ,
259  A_hata_evol[jtime-1], xijm1_a) ;
260 
261  // Parameters for the Dirac systems
262  //---------------------------------
263 
264  Param par_bc_hh ;
265  par_bc_hh.add_int(nz_bound) ;
266  Tbl* cf_b_hh = new Tbl(10) ;
267  cf_b_hh->annule_hard() ;
268  cf_b_hh->set(0) = 11*Rmax + 12*pdt ; // mu
269  cf_b_hh->set(1) = 6*Rmax*pdt ; // d mu / dr
270  cf_b_hh->set(2) = 0 ; // X
271  cf_b_hh->set(3) = 0 ; // d X / dr
272  cf_b_hh->set(4) = 11*Rmax*Rmax + 18*Rmax*pdt ; // h^rr
273  cf_b_hh->set(5) = 6*Rmax*Rmax*pdt ; // d h^rr / dr
274  cf_b_hh->set(6) = 0 ; //eta
275  cf_b_hh->set(7) = 0 ; //d eta / dr
276  cf_b_hh->set(8) = 0 ; //W
277  cf_b_hh->set(9) = 0 ; //d W / dr
278  par_bc_hh.add_tbl_mod(*cf_b_hh, 0) ;
279  Tbl* kib_hh = new Tbl(np2, nt) ;
280  Tbl& khib_hh = *kib_hh ;
281  khib_hh.annule_hard() ;
282  par_bc_hh.add_tbl_mod(khib_hh,1) ;
283  Tbl* mb_hh = new Tbl(np2, nt) ;
284  Tbl& mub_hh = *mb_hh ;
285  mub_hh.annule_hard() ;
286  par_bc_hh.add_tbl_mod(mub_hh, 2) ;
287 
288  Param par_mat_hh ;
289 
290  Tbl xij_mu_hh(np2, nt) ;
291  xij_mu_hh.set_etat_qcq() ;
292  initialize_outgoing_BC(nz_bound, mu_hh_evol[jtime] , mu_a_evol[jtime], xij_mu_hh) ;
293  Tbl xijm1_mu_hh(np2, nt) ;
294  xijm1_mu_hh.set_etat_qcq() ;
295  initialize_outgoing_BC(nz_bound, mu_hh_evol[jtime-1] , mu_a_evol[jtime-1],
296  xijm1_mu_hh) ;
297 
298  Tbl xij_ki_hh(np2, nt) ;
299  xij_ki_hh.set_etat_qcq() ;
300  initialize_outgoing_BC(nz_bound, khi_hh_evol[jtime] , khi_a_evol[jtime], xij_ki_hh) ;
301  Tbl xijm1_ki_hh(np2, nt) ;
302  xijm1_ki_hh.set_etat_qcq() ;
303  initialize_outgoing_BC(nz_bound, khi_hh_evol[jtime-1] , khi_a_evol[jtime-1],
304  xijm1_ki_hh) ;
305 
306  Param par_bc_hata ;
307  par_bc_hata.add_int(nz_bound) ;
308  Tbl* cf_b_hata = new Tbl(10) ;
309  cf_b_hata->annule_hard() ;
310  cf_b_hata->set(0) = 11*Rmax + 12*pdt ; // mu
311  cf_b_hata->set(1) = 6*Rmax*pdt ; // d mu / dr
312  cf_b_hata->set(2) = 0 ; // X
313  cf_b_hata->set(3) = 0 ; // d X / dr
314  cf_b_hata->set(4) = 11*Rmax*Rmax + 18*Rmax*pdt ; // h^rr
315  cf_b_hata->set(5) = 6*Rmax*Rmax*pdt ; // d h^rr / dr
316  cf_b_hata->set(6) = 0 ; //eta
317  cf_b_hata->set(7) = 0 ; //d eta / dr
318  cf_b_hata->set(8) = 0 ; //W
319  cf_b_hata->set(9) = 0 ; //d W / dr
320  par_bc_hata.add_tbl_mod(*cf_b_hata, 0) ;
321  Tbl* kib_hata = new Tbl(np2, nt) ;
322  Tbl& khib_hata = *kib_hata ;
323  khib_hata.annule_hard() ;
324  par_bc_hata.add_tbl_mod(khib_hata,1) ;
325  Tbl* mb_hata = new Tbl(np2, nt) ;
326  Tbl& mub_hata = *mb_hata ;
327  mub_hata.annule_hard() ;
328  par_bc_hata.add_tbl_mod(mub_hata, 2) ;
329 
330  Param par_mat_hata ;
331 
332  Tbl xij_mu_a(np2, nt) ;
333  xij_mu_a.set_etat_qcq() ;
334  initialize_outgoing_BC(nz_bound, mu_a_evol[jtime] ,
335  mu_a_evol.time_derive(jtime, 2), xij_mu_a) ;
336  Tbl xijm1_mu_a(np2, nt) ;
337  xijm1_mu_a.set_etat_qcq() ;
338  tmp = ( mu_a_evol[jtime] - mu_a_evol[jtime-2] ) / (2.*pdt) ;
339  initialize_outgoing_BC(nz_bound, mu_a_evol[jtime-1] , tmp, xijm1_mu_a) ;
340 
341  Tbl xij_ki_a(np2, nt) ;
342  xij_ki_a.set_etat_qcq() ;
343  initialize_outgoing_BC(nz_bound, khi_a_evol[jtime] ,
344  khi_a_evol.time_derive(jtime, 2), xij_ki_a) ;
345  Tbl xijm1_ki_a(np2, nt) ;
346  xijm1_ki_a.set_etat_qcq() ;
347  tmp = ( khi_a_evol[jtime] - khi_a_evol[jtime-2] ) / (2.*pdt) ;
348  initialize_outgoing_BC(nz_bound, khi_a_evol[jtime-1] , tmp, xijm1_ki_a) ;
349 
350  // Quantities at new time-step
351  //----------------------------
352  Scalar n_new(map) ;
353  Scalar psi_new(map) ;
354  Scalar npsi_new(map) ;
355  Vector beta_new(map, CON, triad) ;
356  Scalar A_hh_new(map) ;
357  Scalar B_hh_new(map) ;
358  Scalar A_hata_new(map) ;
359  Scalar B_hata_new(map) ;
360 
361  // Successive values of various quantities:
362  // ---------------------------------------
364  Evolution_full<double> test_ham_constr ;
365  Evolution_full<double> test_mom_constr_r ;
366  Evolution_full<double> test_mom_constr_t ;
367  Evolution_full<double> test_mom_constr_p ;
368  Evolution_full<Tbl> nn_monitor ;
369  Evolution_full<Tbl> psi_monitor ;
370  Evolution_full<Tbl> A_h_monitor ;
371  Evolution_full<Tbl> B_h_monitor ;
372  Evolution_full<Tbl> trh_monitor ;
373  Evolution_full<Tbl> beta_monitor_maxabs ;
374  Evolution_full<Tbl> hh_monitor_central ;
375  Evolution_full<Tbl> hh_monitor_maxabs ;
376  Evolution_full<Tbl> hata_monitor_central ;
377  Evolution_full<Tbl> hata_monitor_maxabs ;
378  Evolution_full<Tbl> check_evol ;
379  Tbl select_scalar(6) ;
380  Tbl select_tens(6) ;
381 
382  Vector zero_vec( map, CON, map.get_bvect_spher() ) ;
383  zero_vec.set_etat_zero() ;
384  const Vector& hat_S = ( mom_euler == 0x0 ? zero_vec : *mom_euler ) ;
385  Scalar lapB(map) ;
386  Scalar lapBm1 = source_B_hata_evol[jtime-1] ;
387  Scalar lapBm2 = source_B_hata_evol[jtime-2] ;
388 
389  // Evolution loop
390  // --------------
391 
392  for (int jt = 0; jt < nb_time_steps; jt++) {
393 
394  double ttime = the_time[jtime] ;
395  k_dd() ;
396 
397  if (jt%check_mod == 0) {
398  cout <<
399  "==============================================================\n"
400  << " step: " << jtime << " time = " << the_time[jtime] << endl
401  << " ADM mass : " << adm_mass()
402  << ", Log of central lapse: " << log(nn().val_grid_point(0,0,0,0)) << endl
403  << "==============================================================\n" ;
404 
405  // Monitoring
406  // ----------
407  m_adm.update(adm_mass(), jtime, the_time[jtime]) ;
408  if (jt > 0) des_evol(m_adm, "ADM mass", "Variation of ADM mass",
409  ngraph0_mon, graph_device) ;
410 
411 
412  nn_monitor.update(monitor_scalar(nn(), select_scalar),
413  jtime, the_time[jtime]) ;
414 
415  psi_monitor.update(monitor_scalar(psi(), select_scalar),
416  jtime, the_time[jtime]) ;
417 
418  A_h_monitor.update(monitor_scalar(A_hh(), select_scalar),
419  jtime, the_time[jtime]) ;
420 
421  B_h_monitor.update(monitor_scalar(B_hh(), select_scalar),
422  jtime, the_time[jtime]) ;
423 
424  trh_monitor.update(monitor_scalar(trh(), select_scalar),
425  jtime, the_time[jtime]) ;
426 
427  beta_monitor_maxabs.update(maxabs_all_domains(beta(), -1, 0x0, cout, verbose),
428  jtime, the_time[jtime]) ;
429 
430  hh_monitor_central.update(central_value(hh()),
431  jtime, the_time[jtime]) ;
432 
433  hh_monitor_maxabs.update(maxabs_all_domains(hh(), -1, 0x0, cout, verbose),
434  jtime, the_time[jtime]) ;
435 
436  hata_monitor_central.update(central_value(hata()),
437  jtime, the_time[jtime]) ;
438 
439  hata_monitor_maxabs.update(maxabs_all_domains(hata(), -1, 0x0, cout, verbose),
440  jtime, the_time[jtime]) ;
441 
442 
443  int jt_graph = jt / check_mod ;
444 
445  Tbl tham = check_hamiltonian_constraint(0x0, cout, verbose) ;
446  double max_error = tham(0,0) ;
447  for (int l=1; l<nz-1; l++) { // all domains but the last one
448  double xx = fabs(tham(0,l)) ;
449  if (xx > max_error) max_error = xx ;
450  }
451  test_ham_constr.update(max_error, jt_graph, the_time[jtime]) ;
452  if (jt > 0) des_evol(test_ham_constr, "Absolute error",
453  "Check of Hamiltonian constraint",
454  ngraph0_mon+1, graph_device) ;
455 
456  Tbl tmom = check_momentum_constraint(0x0, cout, verbose) ;
457  max_error = tmom(0,0) ;
458  for (int l=1; l<nz-1; l++) { // all domains but the last one
459  double xx = fabs(tmom(0,l)) ;
460  if (xx > max_error) max_error = xx ;
461  }
462  test_mom_constr_r.update(max_error, jt_graph, the_time[jtime]) ;
463  if (jt > 0) des_evol(test_mom_constr_r, "Absolute error",
464  "Check of momentum constraint (r comp.)", ngraph0_mon+2,
465  graph_device) ;
466 
467  max_error = tmom(1,0) ;
468  for (int l=1; l<nz-1; l++) { // all domains but the last one
469  double xx = fabs(tmom(1,l)) ;
470  if (xx > max_error) max_error = xx ;
471  }
472  test_mom_constr_t.update(max_error, jt_graph, the_time[jtime]) ;
473  if (jt > 0) des_evol(test_mom_constr_t, "Absolute error",
474  "Check of momentum constraint (\\gh comp.)", ngraph0_mon+3,
475  graph_device) ;
476 
477  max_error = tmom(2,0) ;
478  for (int l=1; l<nz-1; l++) { // all domains but the last one
479  double xx = fabs(tmom(2,l)) ;
480  if (xx > max_error) max_error = xx ;
481  }
482  test_mom_constr_p.update(max_error, jt_graph, the_time[jtime]) ;
483  if (jt > 0) des_evol(test_mom_constr_p, "Absolute error",
484  "Check of momentum constraint (\\gf comp.)", ngraph0_mon+4,
485  graph_device) ;
486 
487  if (jt>2) {
488  Tbl tevol = check_dynamical_equations(0x0, 0x0, cout, verbose) ;
489  Tbl evol_check(6) ; evol_check.set_etat_qcq() ;
490  for (int i=1; i<=3; i++)
491  for(int j=1; j<=i; j++) {
492  max_error = tevol(i, j, 0) ;
493  for (int l=1; l<nz-1; l++) {
494  double xx = fabs(tevol(i,j,l)) ;
495  if (xx > max_error) max_error = xx ;
496  }
497  evol_check.set(i) = max_error ;
498  }
499  check_evol.update(evol_check, jtime, the_time[jtime]) ;
500  }
501  }
502 
503  if (jt%save_mod == 0) {
504  m_adm.save("adm_mass.d") ;
505  nn_monitor.save("nn_monitor.d") ;
506  psi_monitor.save("psi_monitor.d") ;
507  A_h_monitor.save("potA_monitor.d") ;
508  B_h_monitor.save("potB_monitor.d") ;
509  trh_monitor.save("trh_monitor.d") ;
510  beta_monitor_maxabs.save("beta_monitor_maxabs.d") ;
511  hh_monitor_central.save("hh_monitor_central.d") ;
512  hh_monitor_maxabs.save("hh_monitor_maxabs.d") ;
513  hata_monitor_central.save("hata_monitor_central.d") ;
514  hata_monitor_maxabs.save("hata_monitor_maxabs.d") ;
515  test_ham_constr.save("test_ham_constr.d") ;
516  test_mom_constr_r.save("test_mom_constr_r.d") ;
517  test_mom_constr_t.save("test_mom_constr_t.d") ;
518  test_mom_constr_p.save("test_mom_constr_p.d") ;
519  check_evol.save("evol_equations.d") ;
520 
521  save("sigma") ;
522 
523  }
524 
525 
526  // Resolution of hyperbolic equations
527  // ----------------------------------
528  compute_sources(strain_euler) ;
529 
530  A_hata_new = A_hata_evol[jtime]
531  + pdt*( an*source_A_hata_evol[jtime] + anm1*source_A_hata_evol[jtime-1]
532  + anm2*source_A_hata_evol[jtime-2] ) ;
533  B_hata_new = B_hata_evol[jtime]
534  + pdt*( an*source_B_hata_evol[jtime] + anm1*source_B_hata_evol[jtime-1]
535  + anm2*source_B_hata_evol[jtime-2] ) ;
536 
537  A_hh_new = A_hh_evol[jtime]
538  + pdt*( an*source_A_hh_evol[jtime] + anm1*source_A_hh_evol[jtime-1]
539  + anm2*source_A_hh_evol[jtime-2] ) ;
540 
541  B_hh_new = B_hh_evol[jtime]
542  + pdt*( an*source_B_hh_evol[jtime] + anm1*source_B_hh_evol[jtime-1]
543  + anm2*source_B_hh_evol[jtime-2] ) ;
544 
545  Scalar bc_A = -2.*A_hata_new ;
546  bc_A.set_spectral_va().ylm() ;
547  evolve_outgoing_BC(pdt, nz_bound, A_hh_evol[jtime], bc_A, xij_a, xijm1_a,
548  Acc, 0) ;
549  A_hh_new.match_tau(par_A, &par_mat_A_hh) ;
550 
551  Scalar bc_B = -2.*B_hata_new ;
552  bc_B.set_spectral_va().ylm() ;
553  evolve_outgoing_BC(pdt, nz_bound, B_hh_evol[jtime], bc_B, xij_b, xijm1_b,
554  Bcc, -1) ;
555  B_hh_new.match_tau(par_B, &par_mat_B_hh) ;
556 
557  // Boundary conditions for hh and hata
558  //------------------------------------
559  Scalar sbcmu = (18*mu_hh_evol[jtime] - 9*mu_hh_evol[jtime-1]
560  + 2*mu_hh_evol[jtime-2]) / (6*pdt) ;
561  if (sbcmu.get_etat() == ETATZERO) {
562  sbcmu.annule_hard() ;
563  sbcmu.set_spectral_base(base_pseudo) ;
564  }
565  sbcmu.set_spectral_va().ylm() ;
566  tmp = mu_hh_evol[jtime] ;
567  if (tmp.get_etat() == ETATZERO) {
568  tmp.annule_hard() ;
569  tmp.set_spectral_base(base_pseudo) ;
570  }
571  tmp.set_spectral_va().ylm() ;
572  evolve_outgoing_BC(pdt, nz_bound, tmp, sbcmu, xij_mu_hh, xijm1_mu_hh,
573  mub_hh, 0) ;
574  mub_hh *= 6*pdt ;
575 
576  Scalar sbckhi = (18*khi_hh_evol[jtime] - 9*khi_hh_evol[jtime-1]
577  + 2*khi_hh_evol[jtime-2]) / (6*pdt) ;
578  if (sbckhi.get_etat() == ETATZERO) {
579  sbckhi.annule_hard() ;
580  sbckhi.set_spectral_base(base_ref) ;
581  }
582  sbckhi.set_spectral_va().ylm() ;
583  tmp = khi_hh_evol[jtime] ;
584  if (tmp.get_etat() == ETATZERO) {
585  tmp.annule_hard() ;
586  tmp.set_spectral_base(base_ref) ;
587  }
588  tmp.set_spectral_va().ylm() ;
589  evolve_outgoing_BC(pdt, nz_bound, tmp, sbckhi, xij_ki_hh, xijm1_ki_hh,
590  khib_hh, 0) ;
591  khib_hh *= 6*pdt ;
592 
593  sbcmu = (18*mu_a_evol[jtime] - 9*mu_a_evol[jtime-1]
594  + 2*mu_a_evol[jtime-2]) / (6*pdt) ;
595  if (sbcmu.get_etat() == ETATZERO) {
596  sbcmu.annule_hard() ;
597  sbcmu.set_spectral_base(base_pseudo) ;
598  }
599  sbcmu.set_spectral_va().ylm() ;
600  tmp = mu_a_evol[jtime] ;
601  if (tmp.get_etat() == ETATZERO) {
602  tmp.annule_hard() ;
603  tmp.set_spectral_base(base_pseudo) ;
604  }
605  tmp.set_spectral_va().ylm() ;
606  evolve_outgoing_BC(pdt, nz_bound, tmp, sbcmu, xij_mu_a, xijm1_mu_a,
607  mub_hata, 0) ;
608  mub_hata *= 6*pdt ;
609 
610  sbckhi = (18*khi_a_evol[jtime] - 9*khi_a_evol[jtime-1]
611  + 2*khi_a_evol[jtime-2]) / (6*pdt) ;
612  if (sbckhi.get_etat() == ETATZERO) {
613  sbckhi.annule_hard() ;
614  sbckhi.set_spectral_base(base_ref) ;
615  }
616  sbckhi.set_spectral_va().ylm() ;
617  tmp = khi_a_evol[jtime] ;
618  if (tmp.get_etat() == ETATZERO) {
619  tmp.annule_hard() ;
620  tmp.set_spectral_base(base_ref) ;
621  }
622  tmp.set_spectral_va().ylm() ;
623  evolve_outgoing_BC(pdt, nz_bound, tmp, sbckhi, xij_ki_a, xijm1_ki_a,
624  khib_hata, 0) ;
625  khib_hata *= 6*pdt ;
626 
627  // Advance in time
628  // ---------------
629 
630  jtime++ ;
631  ttime += pdt ;
632  the_time.update(ttime, jtime, ttime) ;
633 
634  // Setting As and Bs for h^{ij} and \hat{A}^{ij}
635  set_AB_hata(A_hata_new, B_hata_new) ;
636  set_AB_hh(A_hh_new, B_hh_new) ;
637 
638  hij_tt.set_A_tildeB( A_hh_new, B_hh_new, &par_bc_hh, &par_mat_hh ) ;
639  for (int i=1; i<=3; i++)
640  for (int j=i; j<=3; j++)
641  for (int l=nz_bound+1; l<nz; l++)
642  hij_tt.set(i,j).set_domain(l) = hijtt_old(i,j).domain(l) ;
643  hata_tt.set_A_tildeB( A_hata_new, B_hata_new, &par_bc_hata, &par_mat_hata ) ;
644  for (int i=1; i<=3; i++)
645  for (int j=i; j<=3; j++) {
646  for (int l=nz_bound+1; l<nz; l++)
647  hata_tt.set(i,j).set_domain(l) = hatatt_old(i,j).domain(l) ;
648  hata_tt.set(i,j).set_dzpuis(2) ;
649  }
650 
651  // Computation of h^{ij} at new time-step
652  hh_det_one(hij_tt, &par_mat_hh) ;
653 
654  // Reset of derived quantities
655  del_deriv() ;
656 
657  // Update of khi's and mu's
658  //-------------------------
659  tmp = hij_tt( 1, 1 ) ;
660  tmp.mult_r() ; tmp.mult_r() ;
661  khi_hh_evol.update( tmp, jtime, the_time[jtime] ) ;
662  mu_hh_evol.update( hij_tt.mu(), jtime, the_time[jtime] ) ;
663  tmp = hata_tt( 1, 1 ) ;
664  tmp.mult_r() ; tmp.mult_r() ;
665  khi_a_evol.update( tmp, jtime, the_time[jtime] ) ;
666  mu_a_evol.update( hata_tt.mu(), jtime, the_time[jtime] ) ;
667 
668  // Resolution of elliptic equations
669  // --------------------------------
670  psi_evol.update(psi_evol[jtime-1], jtime, ttime) ;
671 
672  // \hat{A}^{ij} is computed at the new time-step
673  compute_X_from_momentum_constraint(hat_S, hata_tt, niter_elliptic) ;
674 
675  // Iteration on the conformal factor
676  for (int k = 0; k < niter_elliptic; k++) {
677 
678  psi_new = solve_psi(ener_euler) ;
679  psi_new = relax * psi_new + (1.-relax) * psi() ;
680  set_psi_del_npsi(psi_new) ;
681  }
682 
684 
685  // Iteration on N*Psi ## play with the number of iterations...
686  npsi_evol.update(psi_evol[jtime-1], jtime, ttime) ;
687  for (int k = 0; k < niter_elliptic; k++) {
688 
689  npsi_new = solve_npsi( ener_euler, s_euler ) ;
690  npsi_new = relax * npsi_new + (1.-relax) * npsi() ;
691  set_npsi_del_n(npsi_new) ;
692  }
693 
694  // Iteration on beta ## play with the number of iterations...
695  beta_evol.update(beta_evol[jtime-1], jtime, ttime) ;
696  for (int k = 0; k < niter_elliptic; k++) {
697 
698  beta_new = solve_beta(method_poisson_vect) ;
699  beta_new = relax * beta_new + (1.-relax) * beta() ;
700  beta_evol.update(beta_new, jtime, ttime) ;
701  }
702 
703  des_meridian(vec_X()(1), 0., ray_des, "\\gb\\ur\\d", ngraph0+6,
704  graph_device) ;
705  des_meridian(vec_X()(2), 0., ray_des, "\\gb\\u\\gh\\d", ngraph0+7,
706  graph_device) ;
707  des_meridian(vec_X()(3), 0., ray_des, "\\gb\\u\\gf\\d", ngraph0+8,
708  graph_device) ;
709  tmp = A_hh() ;
710  tmp.set_spectral_va().ylm_i() ;
711  des_meridian(tmp, 0., ray_des, "A\\dh", ngraph0+9,
712  graph_device) ;
713  tmp = B_hh_new;
714  tmp.set_spectral_va().ylm_i() ;
715  des_meridian(tmp, 0., ray_des, "B\\dh", ngraph0+10,
716  graph_device) ;
717  des_meridian(trh(), 0., ray_des, "tr h", ngraph0+11,
718  graph_device) ;
719  des_meridian(hh()(1,1), 0., ray_des, "h\\urr\\d", ngraph0+12,
720  graph_device) ;
721  des_meridian(hh()(2,3), 0., ray_des, "h\\u\\gh\\gf\\d", ngraph0+13,
722  graph_device) ;
723  des_meridian(hh()(3,3), 0., ray_des, "h\\u\\gf\\gf\\d", ngraph0+14,
724  graph_device) ;
725 
726  arrete(nopause) ;
727  }
728 
729  par_A.clean_all() ;
730  par_B.clean_all() ;
731  par_mat_A_hh.clean_all() ;
732  par_mat_B_hh.clean_all() ;
733 
734  par_bc_hh.clean_all() ;
735  par_mat_hh.clean_all() ;
736 
737  par_bc_hata.clean_all() ;
738  par_mat_hata.clean_all() ;
739 }
740 
741 
742 //***************************************************************************
743 
744 const Tbl& monitor_scalar(const Scalar& uu, Tbl& resu) {
745 
746  assert( resu.get_ndim() == 1) ;
747  assert( resu.get_taille() >= 6) ;
748 
749  resu.set_etat_qcq() ;
750 
751  resu.set(0) = uu.val_grid_point(0,0,0,0) ;
752  resu.set(1) = max(max(uu)) ;
753  resu.set(2) = min(min(uu)) ;
754 
755  const Mg3d& mg = *(uu.get_mp().get_mg()) ;
756 
757  int nz = mg.get_nzone() ;
758  int nzm1 = nz - 1 ;
759  int nr = mg.get_nr(nzm1) ;
760  int nt = mg.get_nt(nzm1) ;
761  int np = mg.get_np(nzm1) ;
762 
763  resu.set(3) = uu.val_grid_point(nzm1, 0, 0, nr-1) ;
764  resu.set(4) = uu.val_grid_point(nzm1, 0, nt-1, nr-1) ;
765  resu.set(5) = uu.val_grid_point(nzm1, np/2, nt-1, nr-1) ;
766 
767  return resu ;
768 }
769 }
Bases of the spectral expansions.
Definition: base_val.h:322
void mult_x()
The basis is transformed as with a multiplication by .
void mult_cost()
The basis is transformed as with a multiplication.
Vectorial bases (triads) with respect to which the tensorial components are defined.
Definition: base_vect.h:105
Time evolution with full storage (*** under development ***).
Definition: evolution.h:270
virtual void update(const TyT &new_value, int j, double time_j)
Sets a new value at a given time step.
Time evolution with partial storage (*** under development ***).
Definition: evolution.h:371
virtual void update(const TyT &new_value, int j, double time_j)
Sets a new value at a given time step.
void save(const char *filename) const
Saves *this in a formatted file.
Definition: evolution.C:589
TyT time_derive(int j, int n=2) const
Computes the time derivative at time step j by means of a n-th order scheme, from the values at steps...
Definition: evolution.C:504
Base class for coordinate mappings.
Definition: map.h:670
const Mg3d * get_mg() const
Gives the Mg3d on which the mapping is defined.
Definition: map.h:765
const Base_vect_spher & get_bvect_spher() const
Returns the orthonormal vectorial basis associated with the coordinates of the mapping.
Definition: map.h:783
virtual double val_r(int l, double xi, double theta, double pphi) const =0
Returns the value of the radial coordinate r for a given in a given domain.
Multi-domain grid.
Definition: grilles.h:273
int get_np(int l) const
Returns the number of points in the azimuthal direction ( ) in domain no. l.
Definition: grilles.h:462
int get_nt(int l) const
Returns the number of points in the co-latitude direction ( ) in domain no. l.
Definition: grilles.h:457
int get_nzone() const
Returns the number of domains.
Definition: grilles.h:448
int get_nr(int l) const
Returns the number of points in the radial direction ( ) in domain no. l.
Definition: grilles.h:452
Parameter storage.
Definition: param.h:125
void clean_all()
Deletes all the objects stored as modifiables, i.e.
Definition: param.C:174
void add_double_mod(double &x, int position=0)
Adds the address of a new modifiable double to the list.
Definition: param.C:453
void add_tbl_mod(Tbl &ti, int position=0)
Adds the address of a new modifiable Tbl to the list.
Definition: param.C:591
void add_int(const int &n, int position=0)
Adds the address of a new int to the list.
Definition: param.C:246
Tensor field of valence 0 (or component of a tensorial field).
Definition: scalar.h:387
const Base_val & get_spectral_base() const
Returns the spectral bases of the Valeur va
Definition: scalar.h:1294
Tbl & set_domain(int l)
Read/write of the value in a given domain.
Definition: scalar.h:615
virtual void std_spectral_base()
Sets the spectral bases of the Valeur va to the standard ones for a scalar field.
Definition: scalar.C:784
double val_grid_point(int l, int k, int j, int i) const
Returns the value of the field at a specified grid point.
Definition: scalar.h:637
void annule_hard()
Sets the Scalar to zero in a hard way.
Definition: scalar.C:380
int get_etat() const
Returns the logical state ETATNONDEF (undefined), ETATZERO (null) or ETATQCQ (ordinary).
Definition: scalar.h:554
void set_dzpuis(int)
Modifies the dzpuis flag.
Definition: scalar.C:808
Valeur & set_spectral_va()
Returns va (read/write version)
Definition: scalar.h:604
void mult_r()
Multiplication by r everywhere; dzpuis is not changed.
void set_spectral_base(const Base_val &)
Sets the spectral bases of the Valeur va
Definition: scalar.C:797
Transverse symmetric tensors of rank 2.
Definition: sym_tensor.h:608
Transverse and traceless symmetric tensors of rank 2.
Definition: sym_tensor.h:938
void set_A_tildeB(const Scalar &a_in, const Scalar &tb_in, Param *par_bc=0x0, Param *par_mat=0x0)
Assigns the derived members A and .
Class intended to describe valence-2 symmetric tensors.
Definition: sym_tensor.h:223
const Scalar & mu(Param *par=0x0) const
Gives the field (see member p_mu ).
Basic array class.
Definition: tbl.h:161
int get_ndim() const
Gives the number of dimensions (ie dim.ndim)
Definition: tbl.h:400
void annule_hard()
Sets the Tbl to zero in a hard way.
Definition: tbl.C:372
double & set(int i)
Read/write of a particular element (index i) (1D case)
Definition: tbl.h:281
void set_etat_qcq()
Sets the logical state to ETATQCQ (ordinary state).
Definition: tbl.C:361
int get_taille() const
Gives the total size (ie dim.taille)
Definition: tbl.h:397
virtual const Scalar & npsi() const
Factor at the current time step (jtime ).
Evolution_std< Sym_tensor > hh_evol
Values at successive time steps of the components .
Definition: time_slice.h:530
virtual const Sym_tensor & k_dd() const
Extrinsic curvature tensor (covariant components ) at the current time step (jtime )
virtual const Scalar & nn() const
Lapse function N at the current time step (jtime )
virtual void set_psi_del_npsi(const Scalar &psi_in)
Sets the conformal factor relating the physical metric to the conformal one: .
virtual const Sym_tensor & hata() const
Conformal representation of the traceless part of the extrinsic curvature: .
Evolution_std< Sym_tensor > hata_evol
Values at successive time steps of the components .
Definition: time_slice.h:542
virtual const Vector & vec_X(int method_poisson=6) const
Vector representing the longitudinal part of .
virtual const Scalar & psi() const
Conformal factor relating the physical metric to the conformal one: .
virtual void del_deriv() const
Deletes all the derived quantities.
Evolution_std< Scalar > npsi_evol
Values at successive time steps of the factor .
Definition: time_slice.h:522
virtual void set_npsi_del_n(const Scalar &npsi_in)
Sets the factor at the current time step (jtime ) and deletes the value of N.
Evolution_std< Scalar > psi_evol
Values at successive time steps of the conformal factor relating the physical metric to the conform...
Definition: time_slice.h:517
Evolution_std< Scalar > A_hata_evol
Potential A associated with the symmetric tensor .
Definition: time_slice.h:547
void compute_X_from_momentum_constraint(const Vector &hat_S, const Sym_tensor_tt &hata_tt, int iter_max=200, double precis=1.e-12, double relax=0.8, int methode_poisson=6)
Computes the vector from the conformally-rescaled momentum , using the momentum constraint.
Evolution_std< Scalar > B_hata_evol
Potential associated with the symmetric tensor .
Definition: time_slice.h:552
int jtime
Time step index of the latest slice.
Definition: time_slice.h:190
void save(const char *rootname) const
Saves in a binary file.
Definition: time_slice.C:461
Tbl check_hamiltonian_constraint(const Scalar *energy_density=0x0, ostream &ost=cout, bool verb=true) const
Checks the level at which the hamiltonian constraint is verified.
virtual const Vector & beta() const
shift vector at the current time step (jtime )
int depth
Number of stored time slices.
Definition: time_slice.h:179
Evolution_std< double > the_time
Time label of each slice.
Definition: time_slice.h:193
Tbl check_dynamical_equations(const Sym_tensor *strain_tensor=0x0, const Scalar *energy_density=0x0, ostream &ost=cout, bool verb=true) const
Checks the level at which the dynamical equations are verified.
Evolution_std< Vector > beta_evol
Values at successive time steps of the shift vector .
Definition: time_slice.h:219
Tbl check_momentum_constraint(const Vector *momentum_density=0x0, ostream &ost=cout, bool verb=true) const
Checks the level at which the momentum constraints are verified.
virtual const Scalar & trh() const
Computes the trace h, with respect to the flat metric ff , of .
virtual double adm_mass() const
Returns the ADM mass at (geometrical units) the current step.
virtual const Sym_tensor & hh(Param *par_bc=0x0, Param *par_mat=0x0) const
Deviation of the conformal metric from the flat metric : .
virtual Scalar solve_psi(const Scalar *ener_dens=0x0) const
Solves the elliptic equation for the conformal factor $\Psi$ (Hamiltonian constraint).
Evolution_std< Scalar > source_B_hh_evol
The potential of the source of equation for .
Definition: time_slice.h:995
Evolution_std< Scalar > source_A_hh_evol
The A potential of the source of equation for .
Definition: time_slice.h:989
void evolve(double pdt, int nb_time_steps, int niter_elliptic, double relax_elliptic, int check_mod, int save_mod, int method_poisson_vect=6, int nopause=1, const char *graph_device=0x0, bool verbose=true, const Scalar *ener_euler=0x0, const Vector *mom_euler=0x0, const Scalar *s_euler=0x0, const Sym_tensor *strain_euler=0x0)
Time evolution by resolution of Einstein equations.
virtual const Scalar & B_hh() const
Returns the potential of .
Evolution_std< Scalar > A_hh_evol
The A potential of .
Definition: time_slice.h:977
void hh_det_one(int j, Param *par_bc=0x0, Param *par_mat=0x0) const
Computes from the values of A and and using the condition , which fixes the trace of .
Evolution_std< Scalar > source_B_hata_evol
The potential of the source of equation for .
Definition: time_slice.h:1007
void compute_sources(const Sym_tensor *strain_tensor=0x0) const
Computes the sources source_A_XXX_evol and source_B_XXX_evol , for the solution of the evolution equa...
virtual Scalar solve_npsi(const Scalar *ener_dens=0x0, const Scalar *trace_stress=0x0) const
Solves the elliptic equation for (maximal slicing condition + Hamiltonian constraint)
virtual const Scalar & A_hh() const
Returns the potential A of .
virtual Vector solve_beta(int method=6) const
Solves the elliptic equation for the shift vector from (Eq.
virtual void set_AB_hh(const Scalar &A_in, const Scalar &B_in)
Sets the potentials A and of the TT part of (see the documentation of Sym_tensor for details).
Evolution_std< Scalar > B_hh_evol
The potential of .
Definition: time_slice.h:983
Evolution_std< Scalar > source_A_hata_evol
The potential A of the source of equation for .
Definition: time_slice.h:1001
void ylm()
Computes the coefficients of *this.
Definition: valeur_ylm.C:138
void ylm_i()
Inverse of ylm()
Definition: valeur_ylm_i.C:131
Tensor field of valence 1.
Definition: vector.h:188
Tbl min(const Cmp &)
Minimum values of a Cmp in each domain.
Definition: cmp_math.C:458
Tbl max(const Cmp &)
Maximum values of a Cmp in each domain.
Definition: cmp_math.C:435
Cmp log(const Cmp &)
Neperian logarithm.
Definition: cmp_math.C:296
void des_meridian(const Scalar &uu, double r_min, double r_max, const char *nomy, int ngraph, const char *device=0x0, bool closeit=false, bool draw_bound=true)
Draws 5 profiles of a scalar field along various radial axes in two meridional planes and .
void des_evol(const Evolution< double > &uu, const char *nomy=0x0, const char *title=0x0, int ngraph=0, const char *device=0x0, bool closeit=false, bool show_time=true, const char *nomx=0x0)
Plots the variation of some quantity against time.
void arrete(int a=0)
Setting a stop point in a code.
Definition: arrete.C:61
const Map & get_mp() const
Returns the mapping.
Definition: tensor.h:861
virtual void annule(int l_min, int l_max)
Sets the Tensor to zero in several domains.
Definition: tensor.C:671
virtual void inc_dzpuis(int inc=1)
Increases by inc units the value of dzpuis and changes accordingly the values in the compactified ext...
Definition: tensor.C:816
virtual void set_etat_zero()
Sets the logical state of all components to ETATZERO (zero state).
Definition: tensor.C:497
Scalar & set(const Itbl &ind)
Returns the value of a component (read/write version).
Definition: tensor.C:654
const Base_vect * get_triad() const
Returns the vectorial basis (triad) on which the components are defined.
Definition: tensor.h:866
Tbl maxabs_all_domains(const Tensor &aa, int l_excluded=-1, const char *comment=0x0, ostream &ost=cout, bool verb=true)
Maximum of the absolute value of each component of a tensor over all the domains.
Tbl central_value(const Tensor &aa, const char *comment=0x0, ostream &ost=cout)
Central value of each component of a tensor.
Lorene prototypes.
Definition: app_hor.h:64