LORENE
et_bin_bhns_extr_velpot.C
1 /*
2  * Method of class Et_bin_bhns_extr to compute the velocity scalar
3  * potential $\psi$ by solving the continuity equation
4  * in the Kerr-Schild background metric or in the conformally flat one
5  * with extreme mass ratio
6  *
7  * (see file et_bin_bhns_extr.h for documentation).
8  *
9  */
10 
11 /*
12  * Copyright (c) 2004-2005 Keisuke Taniguchi
13  *
14  * This file is part of LORENE.
15  *
16  * LORENE is free software; you can redistribute it and/or modify
17  * it under the terms of the GNU General Public License version 2
18  * as published by the Free Software Foundation.
19  *
20  * LORENE is distributed in the hope that it will be useful,
21  * but WITHOUT ANY WARRANTY; without even the implied warranty of
22  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
23  * GNU General Public License for more details.
24  *
25  * You should have received a copy of the GNU General Public License
26  * along with LORENE; if not, write to the Free Software
27  * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
28  *
29  */
30 
31 
32 
33 /*
34  * $Id: et_bin_bhns_extr_velpot.C,v 1.4 2016/12/05 16:17:52 j_novak Exp $
35  * $Log: et_bin_bhns_extr_velpot.C,v $
36  * Revision 1.4 2016/12/05 16:17:52 j_novak
37  * Suppression of some global variables (file names, loch, ...) to prevent redefinitions
38  *
39  * Revision 1.3 2014/10/13 08:52:55 j_novak
40  * Lorene classes and functions now belong to the namespace Lorene.
41  *
42  * Revision 1.2 2005/02/28 23:17:18 k_taniguchi
43  * Modification to include the case of the conformally flat background metric
44  *
45  * Revision 1.1 2004/11/30 20:51:56 k_taniguchi
46  * *** empty log message ***
47  *
48  *
49  * $Header: /cvsroot/Lorene/C++/Source/Etoile/et_bin_bhns_extr_velpot.C,v 1.4 2016/12/05 16:17:52 j_novak Exp $
50  *
51  */
52 
53 // C headers
54 //#include <math.h>
55 
56 // Lorene headers
57 #include "et_bin_bhns_extr.h"
58 #include "scalar.h"
59 #include "metrique.h"
60 #include "etoile.h"
61 #include "eos.h"
62 #include "param.h"
63 #include "coord.h"
64 #include "unites.h"
65 
66 // Local prototype
67 namespace Lorene {
68 Cmp raccord_c1(const Cmp& uu, int l1) ;
69 
70 double Et_bin_bhns_extr::velocity_pot_extr(const double& mass,
71  const double& sepa,
72  int mermax, double precis,
73  double relax) {
74 
75  using namespace Unites ;
76 
77  if (kerrschild) {
78 
79  if (eos.identify() == 5 || eos.identify() == 4 ||
80  eos.identify() == 3) {
81 
82  cout << "Etoile_bin::velocity_pot_extr" << endl ;
83  cout << "The code has not prepared for this kind of EOS!!!"
84  << endl;
85  abort() ;
86 
87  } // End of strange stars case
88  else {
89 
90  int nzm1 = mp.get_mg()->get_nzone() - 1 ;
91 
92  //--------------------------------
93  // Specific relativistic enthalpy ---> hhh
94  //--------------------------------
95 
96  Tenseur hhh = exp(unsurc2 * ent) ; // = 1 at the Newtonian limit
97  hhh.set_std_base() ;
98 
99  //--------------------------------------------
100  // Computation of W^i = - A^2 h Gamma_n B^i/N
101  //--------------------------------------------
102 
103  Tenseur www = - a_car * hhh * gam_euler * bsn ;
104 
105  www.change_triad( mp.get_bvect_cart() ) ;
106  // components on the mapping Cartesian basis
107 
108  //-------------------------------------------------
109  // Constant value of W^i at the center of the star
110  //-------------------------------------------------
111 
112  Tenseur v_orb(mp, 1, CON, mp.get_bvect_cart()) ;
113 
114  v_orb.set_etat_qcq() ;
115  for (int i=0; i<3; i++) {
116  v_orb.set(i) = www(i)(0, 0, 0, 0) ;
117  }
118 
119  v_orb.set_triad( *(www.get_triad()) ) ;
120  v_orb.set_std_base() ;
121 
122  // Some auxiliary terms
123  // --------------------
124 
125  const Coord& xx = mp.x ;
126  const Coord& yy = mp.y ;
127  const Coord& zz = mp.z ;
128 
129  Tenseur r_bh(mp) ;
130  r_bh.set_etat_qcq() ;
131  r_bh.set() = pow( (xx+sepa)*(xx+sepa) + yy*yy + zz*zz, 0.5) ;
132  r_bh.set_std_base() ;
133 
134  Tenseur msr(mp) ;
135  msr = ggrav * mass / r_bh ;
136  msr.set_std_base() ;
137 
138  Tenseur lapse_bh2(mp) ; // lapse_bh * lapse_bh
139  lapse_bh2 = 1. / (1.+2.*msr) ;
140  lapse_bh2.set_std_base() ;
141 
142  Tenseur lapse_bh3(mp) ; // lapse_bh * lapse_bh * lapse_bh
143  lapse_bh3 = 1./pow(1.+2.*msr, 1.5) ;
144  lapse_bh3.set_std_base() ;
145 
146  Tenseur xx_con(mp, 1, CON, mp.get_bvect_cart()) ;
147  xx_con.set_etat_qcq() ;
148  xx_con.set(0) = xx + sepa ;
149  xx_con.set(1) = yy ;
150  xx_con.set(2) = zz ;
151  xx_con.set_std_base() ;
152 
153  Tenseur xsr_con(mp, 1, CON, mp.get_bvect_cart()) ;
154  xsr_con = xx_con / r_bh ;
155  xsr_con.set_std_base() ;
156 
157  Tenseur xx_cov(mp, 1, COV, mp.get_bvect_cart()) ;
158  xx_cov.set_etat_qcq() ;
159  xx_cov.set(0) = xx + sepa ;
160  xx_cov.set(1) = yy ;
161  xx_cov.set(2) = zz ;
162  xx_cov.set_std_base() ;
163 
164  Tenseur xsr_cov(mp, 1, COV, mp.get_bvect_cart()) ;
165  xsr_cov = xx_cov / r_bh ;
166  xsr_cov.set_std_base() ;
167 
168  // X_i/r_bh in the spherical coordinate
169  const Coord& rr = mp.r ;
170  const Coord& st = mp.sint ;
171  const Coord& ct = mp.cost ;
172  const Coord& sp = mp.sinp ;
173  const Coord& cp = mp.cosp ;
174 
175  Tenseur rr_spher_cov(mp, 1, COV, mp.get_bvect_spher()) ;
176  rr_spher_cov.set_etat_qcq() ;
177  rr_spher_cov.set(0) = rr + sepa*st*cp ;
178  rr_spher_cov.set(1) = sepa*ct*cp ;
179  rr_spher_cov.set(2) = - sepa*sp ;
180  rr_spher_cov.set_std_base() ;
181 
182  Tenseur xsr_spher_cov(mp, 1, COV, mp.get_bvect_spher()) ;
183  xsr_spher_cov = rr_spher_cov / r_bh ;
184  xsr_spher_cov.set_std_base() ;
185 
186  // l^j D_j H_ent
187  Tenseur ldent = flat_scalar_prod(xsr_con, ent.gradient()) ;
188 
189  // l^j D_j beta_auto
190  Tenseur ldbeta = flat_scalar_prod(xsr_con, beta_auto.gradient()) ;
191 
192  // l^j D_j psi0
193  Tenseur ldpsi0 = flat_scalar_prod(xsr_con, psi0.gradient()) ;
194 
195  // l^i D_i (l^j D_j psi0)
196  Tenseur ldldpsi0 = flat_scalar_prod(xsr_con, ldpsi0.gradient()) ;
197 
198  //-------------------------------------------------
199  // Source and coefficients a,b for poisson_compact
200  // (idenpendent from psi0)
201  //-------------------------------------------------
202 
203  Cmp dndh_log = eos.der_nbar_ent(ent(), nzet) ;
204 
205  // In order to avoid any division by zero in the computation of
206  // zeta_h the value of dndh_log is set to 1
207  // in the external domains:
208  for (int l=nzet; l <= nzm1; l++) {
209  dndh_log.set(l) = 1 ;
210  }
211 
212  double erreur ;
213 
214  Tenseur zeta_h( ent() / dndh_log ) ;
215  zeta_h.set_std_base() ;
216 
217  Tenseur tmp_zeta = 1 - unsurc2 * zeta_h ;
218  tmp_zeta.set_std_base() ;
219 
220  Tenseur bb = tmp_zeta * (ent.gradient_spher()
221  - 2.*lapse_bh2 * msr * ldent
222  * xsr_spher_cov)
223  + unsurc2 * zeta_h * (beta_auto.gradient_spher()
224  - 2.*lapse_bh2 * msr * ldbeta
225  * xsr_spher_cov)
226  - unsurc2 * 2. * zeta_h * lapse_bh2 * lapse_bh2 * msr / r_bh
227  * (1.+4.*msr) * xsr_spher_cov ;
228 
229  Tenseur entmb = ent - beta_auto ;
230 
231  Tenseur source = flat_scalar_prod(www - v_orb, ent.gradient())
232  + unsurc2*zeta_h*( flat_scalar_prod(v_orb, entmb.gradient())
234  / gam_euler )
235  + 2.*lapse_bh2 * msr * flat_scalar_prod(xsr_cov, v_orb)
236  * flat_scalar_prod(xsr_con, ent.gradient())
237  + unsurc2 * 2. * zeta_h
238  * (lapse_bh2*msr*(ldldpsi0
239  - flat_scalar_prod(xsr_cov, v_orb)
240  * (flat_scalar_prod(xsr_con, entmb.gradient())
241  - lapse_bh2 * (1.+4.*msr) / r_bh))
242  + a_car * hhh * gam_euler * lapse_bh3 * msr * (1.+3.*msr)
243  / r_bh) ;
244 
245  source.annule(nzet, nzm1) ;
246 
247  //----------------------------------------------------
248  // Resolution by means of Map_radial::poisson_compact
249  //----------------------------------------------------
250 
251  Param par ;
252  int niter ;
253  par.add_int(mermax) ;
254  par.add_double(precis, 0) ;
255  par.add_double(relax, 1) ;
256  par.add_int_mod(niter) ;
257 
258  if (psi0.get_etat() == ETATZERO) {
259  psi0.set_etat_qcq() ;
260  psi0.set() = 0 ;
261  }
262 
263  source.set().va.ylm() ;
264 
265  mp.poisson_compact(source(), zeta_h(), bb, par, psi0.set() ) ;
266 
267  //-----------------------
268  // Check of the solution
269  //-----------------------
270 
271  Tenseur bb_dpsi0 = flat_scalar_prod( bb, psi0.gradient_spher() ) ;
272 
273  Cmp oper = zeta_h() * psi0().laplacien() + bb_dpsi0() ;
274 
275  source.set().va.ylm_i() ;
276 
277  erreur = diffrel(oper, source())(0) ;
278 
279  cout << "Check of the resolution of the continuity equation : "
280  << endl ;
281  cout << "norme(source) : " << norme(source())(0)
282  << " diff oper/source : " << erreur << endl ;
283 
284  //--------------------------
285  // Computation of grad(psi)
286  //--------------------------
287 
288  // The computation is done component by component because
289  // psi0.gradient() is a covariant vector, whereas v_orb is a
290  // contravariant one.
291 
292  d_psi.set_etat_qcq() ;
293 
294  for (int i=0; i<3; i++) {
295  d_psi.set(i) = (psi0.gradient())(i) + v_orb(i) ;
296  }
297 
298  d_psi.set_triad( *(v_orb.get_triad()) ) ;
299 
300  // C^1 continuation of d_psi outside the star
301  // (to ensure a smooth enthalpy field accross the stellar surface)
302  // ----------------------------------------------------------------
303 
304  d_psi.annule(nzet, nzm1) ;
305  for (int i=0; i<3; i++) {
306  d_psi.set(i) = raccord_c1(d_psi(i), nzet) ;
307  }
308 
309  assert( d_psi.get_triad() == &(mp.get_bvect_cart()) ) ;
310 
312 
313  return erreur ;
314 
315  }
316 
317  }
318  else {
319 
320  if (eos.identify() == 5 || eos.identify() == 4 ||
321  eos.identify() == 3) {
322 
323  cout << "Etoile_bin::velocity_pot_extr" << endl ;
324  cout << "The code has not prepared for this kind of EOS!!!"
325  << endl;
326  abort() ;
327 
328  } // End of strange stars case
329  else {
330 
331  int nzm1 = mp.get_mg()->get_nzone() - 1 ;
332 
333  //--------------------------------
334  // Specific relativistic enthalpy ---> hhh
335  //--------------------------------
336 
337  Tenseur hhh = exp(unsurc2 * ent) ; // = 1 at the Newtonian limit
338  hhh.set_std_base() ;
339 
340  //--------------------------------------------
341  // Computation of W^i = - A^2 h Gamma_n B^i/N
342  //--------------------------------------------
343 
344  Tenseur www = - a_car * hhh * gam_euler * bsn ;
345 
346  www.change_triad( mp.get_bvect_cart() ) ;
347  // components on the mapping Cartesian basis
348 
349  //-------------------------------------------------
350  // Constant value of W^i at the center of the star
351  //-------------------------------------------------
352 
353  Tenseur v_orb(mp, 1, CON, mp.get_bvect_cart()) ;
354 
355  v_orb.set_etat_qcq() ;
356  for (int i=0; i<3; i++) {
357  v_orb.set(i) = www(i)(0, 0, 0, 0) ;
358  }
359 
360  v_orb.set_triad( *(www.get_triad()) ) ;
361  v_orb.set_std_base() ;
362 
363  // Some auxiliary terms
364  // --------------------
365 
366  const Coord& xx = mp.x ;
367  const Coord& yy = mp.y ;
368  const Coord& zz = mp.z ;
369 
370  Tenseur r_bh(mp) ;
371  r_bh.set_etat_qcq() ;
372  r_bh.set() = pow( (xx+sepa)*(xx+sepa) + yy*yy + zz*zz, 0.5) ;
373  r_bh.set_std_base() ;
374 
375  Tenseur msr(mp) ;
376  msr = ggrav * mass / r_bh ;
377  msr.set_std_base() ;
378 
379  Tenseur xx_cov(mp, 1, COV, mp.get_bvect_cart()) ;
380  xx_cov.set_etat_qcq() ;
381  xx_cov.set(0) = xx + sepa ;
382  xx_cov.set(1) = yy ;
383  xx_cov.set(2) = zz ;
384  xx_cov.set_std_base() ;
385 
386  // X_i in the spherical coordinate
387  const Coord& rr = mp.r ;
388  const Coord& st = mp.sint ;
389  const Coord& ct = mp.cost ;
390  const Coord& sp = mp.sinp ;
391  const Coord& cp = mp.cosp ;
392 
393  Tenseur rr_spher_cov(mp, 1, COV, mp.get_bvect_spher()) ;
394  rr_spher_cov.set_etat_qcq() ;
395  rr_spher_cov.set(0) = rr + sepa*st*cp ;
396  rr_spher_cov.set(1) = sepa*ct*cp ;
397  rr_spher_cov.set(2) = - sepa*sp ;
398  rr_spher_cov.set_std_base() ;
399 
400  Tenseur tmp_bh(mp) ;
401  tmp_bh = 0.5 * msr * msr / (1.-0.25*msr*msr) / r_bh / r_bh ;
402  tmp_bh.set_std_base() ;
403 
404  //-------------------------------------------------
405  // Source and coefficients a,b for poisson_compact
406  // (idenpendent from psi0)
407  //-------------------------------------------------
408 
409  Cmp dndh_log = eos.der_nbar_ent(ent(), nzet) ;
410 
411  // In order to avoid any division by zero in the computation of
412  // zeta_h the value of dndh_log is set to 1
413  // in the external domains:
414  for (int l=nzet; l <= nzm1; l++) {
415  dndh_log.set(l) = 1 ;
416  }
417 
418  double erreur ;
419 
420  Tenseur zeta_h( ent() / dndh_log ) ;
421  zeta_h.set_std_base() ;
422 
423  Tenseur tmp_zeta = 1 - unsurc2 * zeta_h ;
424  tmp_zeta.set_std_base() ;
425 
426  Tenseur bb = tmp_zeta * ent.gradient_spher()
427  + unsurc2 * zeta_h * (beta_auto.gradient_spher()
428  + tmp_bh * rr_spher_cov) ;
429 
430  Tenseur entmb = ent - beta_auto ;
431 
432  Tenseur source = flat_scalar_prod(www - v_orb, ent.gradient())
433  + unsurc2*zeta_h*( flat_scalar_prod(v_orb, entmb.gradient())
434  - tmp_bh * flat_scalar_prod(v_orb, xx_cov)
436  / gam_euler ) ;
437 
438  source.annule(nzet, nzm1) ;
439 
440  //----------------------------------------------------
441  // Resolution by means of Map_radial::poisson_compact
442  //----------------------------------------------------
443 
444  Param par ;
445  int niter ;
446  par.add_int(mermax) ;
447  par.add_double(precis, 0) ;
448  par.add_double(relax, 1) ;
449  par.add_int_mod(niter) ;
450 
451  if (psi0.get_etat() == ETATZERO) {
452  psi0.set_etat_qcq() ;
453  psi0.set() = 0 ;
454  }
455 
456  source.set().va.ylm() ;
457 
458  mp.poisson_compact(source(), zeta_h(), bb, par, psi0.set() ) ;
459 
460  //-----------------------
461  // Check of the solution
462  //-----------------------
463 
464  Tenseur bb_dpsi0 = flat_scalar_prod( bb, psi0.gradient_spher() ) ;
465 
466  Cmp oper = zeta_h() * psi0().laplacien() + bb_dpsi0() ;
467 
468  source.set().va.ylm_i() ;
469 
470  erreur = diffrel(oper, source())(0) ;
471 
472  cout << "Check of the resolution of the continuity equation : "
473  << endl ;
474  cout << "norme(source) : " << norme(source())(0)
475  << " diff oper/source : " << erreur << endl ;
476 
477  //--------------------------
478  // Computation of grad(psi)
479  //--------------------------
480 
481  // The computation is done component by component because
482  // psi0.gradient() is a covariant vector, whereas v_orb is a
483  // contravariant one.
484 
485  d_psi.set_etat_qcq() ;
486 
487  for (int i=0; i<3; i++) {
488  d_psi.set(i) = (psi0.gradient())(i) + v_orb(i) ;
489  }
490 
491  d_psi.set_triad( *(v_orb.get_triad()) ) ;
492 
493  // C^1 continuation of d_psi outside the star
494  // (to ensure a smooth enthalpy field accross the stellar surface)
495  // ----------------------------------------------------------------
496 
497  d_psi.annule(nzet, nzm1) ;
498  for (int i=0; i<3; i++) {
499  d_psi.set(i) = raccord_c1(d_psi(i), nzet) ;
500  }
501 
502  assert( d_psi.get_triad() == &(mp.get_bvect_cart()) ) ;
503 
505 
506  return erreur ;
507 
508  }
509 
510  }
511 
512 }
513 }
void annule(int l)
Sets the Tenseur to zero in a given domain.
Definition: tenseur.C:916
const Base_vect & ref_triad
Reference triad ("absolute frame"), with respect to which the components of all the member Tenseur &#39;s...
Definition: etoile.h:831
Component of a tensorial field *** DEPRECATED : use class Scalar instead ***.
Definition: cmp.h:446
Cmp exp(const Cmp &)
Exponential.
Definition: cmp_math.C:273
void add_int(const int &n, int position=0)
Adds the address of a new int to the list.
Definition: param.C:249
const Tenseur & gradient_spher() const
Returns the gradient of *this (Spherical coordinates) (scalar field only).
Definition: tenseur.C:1564
void ylm_i()
Inverse of ylm()
Definition: valeur_ylm_i.C:134
void set_triad(const Base_vect &new_triad)
Assigns a new vectorial basis (triad) of decomposition.
Definition: tenseur.C:690
Cmp der_nbar_ent(const Cmp &ent, int nzet, int l_min=0, Param *par=0x0) const
Computes the logarithmic derivative from the log-enthalpy and extra parameters.
Definition: eos.C:431
void set_std_base()
Set the standard spectal basis of decomposition for each component.
Definition: tenseur.C:1186
const Base_vect_spher & get_bvect_spher() const
Returns the orthonormal vectorial basis associated with the coordinates of the mapping.
Definition: map.h:801
Lorene prototypes.
Definition: app_hor.h:67
Standard units of space, time and mass.
void ylm()
Computes the coefficients of *this.
Definition: valeur_ylm.C:141
const Mg3d * get_mg() const
Gives the Mg3d on which the mapping is defined.
Definition: map.h:783
virtual int identify() const =0
Returns a number to identify the sub-classe of Eos the object belongs to.
Tenseur flat_scalar_prod(const Tenseur &t1, const Tenseur &t2)
Scalar product of two Tenseur when the metric is : performs the contraction of the last index of t1 w...
virtual void poisson_compact(const Cmp &source, const Cmp &aa, const Tenseur &bb, const Param &par, Cmp &psi) const =0
Resolution of the elliptic equation in the case where the stellar interior is covered by a single do...
double unsurc2
: unsurc2=1 for a relativistic star, 0 for a Newtonian one.
Definition: etoile.h:445
bool kerrschild
Indicator of the background metric: true for the Kerr-Shild metric, false for the conformally flat on...
void set(const Map *mp, Mtbl *(*construct)(const Map *))
Semi-constructor from a mapping and a method.
Definition: coord.C:137
Tbl diffrel(const Cmp &a, const Cmp &b)
Relative difference between two Cmp (norme version).
Definition: cmp_math.C:507
Coord sint
Definition: map.h:739
Tenseur psi0
Scalar potential of the non-translational part of fluid 4-velocity (in the irrotational case) ...
Definition: etoile.h:836
Cmp & set()
Read/write for a scalar (see also operator=(const Cmp&) ).
Definition: tenseur.C:840
void change_triad(const Base_vect &new_triad)
Sets a new vectorial basis (triad) of decomposition and modifies the components accordingly.
Definition: tenseur.C:684
Tbl norme(const Cmp &)
Sums of the absolute values of all the values of the Cmp in each domain.
Definition: cmp_math.C:484
double velocity_pot_extr(const double &mass, const double &sepa, int mermax, double precis, double relax)
Computes the non-translational part of the velocity scalar potential by solving the continuity equat...
Tenseur bsn
3-vector shift, divided by N , of the rotating coordinates, .
Definition: etoile.h:953
const Base_vect * get_triad() const
Returns the vectorial basis (triad) on which the components are defined.
Definition: tenseur.h:707
Tenseur gam_euler
Lorentz factor between the fluid and Eulerian observers.
Definition: etoile.h:474
Parameter storage.
Definition: param.h:125
Map & mp
Mapping associated with the star.
Definition: etoile.h:432
Coord sinp
Definition: map.h:741
int get_nzone() const
Returns the number of domains.
Definition: grilles.h:465
const Eos & eos
Equation of state of the stellar matter.
Definition: etoile.h:454
int get_etat() const
Returns the logical state.
Definition: tenseur.h:710
Cmp pow(const Cmp &, int)
Power .
Definition: cmp_math.C:351
Active physical coordinates and mapping derivatives.
Definition: coord.h:90
int nzet
Number of domains of *mp occupied by the star.
Definition: etoile.h:435
Tenseur a_car
Total conformal factor .
Definition: etoile.h:518
Tbl & set(int l)
Read/write of the value in a given domain.
Definition: cmp.h:724
const Base_vect_cart & get_bvect_cart() const
Returns the Cartesian basis associated with the coordinates (x,y,z) of the mapping, i.e.
Definition: map.h:809
Tenseur ent
Log-enthalpy (relativistic case) or specific enthalpy (Newtonian case)
Definition: etoile.h:460
Coord y
y coordinate centered on the grid
Definition: map.h:745
Coord cosp
Definition: map.h:742
void add_double(const double &x, int position=0)
Adds the the address of a new double to the list.
Definition: param.C:318
Coord x
x coordinate centered on the grid
Definition: map.h:744
Tenseur beta_auto
Logarithm of the part of the product AN generated principaly by by the star.
Definition: etoile.h:509
void set_etat_qcq()
Sets the logical state to ETATQCQ (ordinary state).
Definition: tenseur.C:652
Coord z
z coordinate centered on the grid
Definition: map.h:746
Valeur va
The numerical value of the Cmp.
Definition: cmp.h:464
Tensor handling *** DEPRECATED : use class Tensor instead ***.
Definition: tenseur.h:304
void add_int_mod(int &n, int position=0)
Adds the address of a new modifiable int to the list.
Definition: param.C:388
const Tenseur & gradient() const
Returns the gradient of *this (Cartesian coordinates)
Definition: tenseur.C:1558
Coord r
r coordinate centered on the grid
Definition: map.h:736
Tenseur d_psi
Gradient of (in the irrotational case) (Cartesian components with respect to ref_triad ) ...
Definition: etoile.h:841
Coord cost
Definition: map.h:740