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#include<math.h>
// Lorene headers
#include "metric.h"
#include "nbr_spx.h"
#include "utilitaires.h"
#include "graphique.h"
#include "proto.h"
#include "diff.h"
namespace Lorene {
// Inversion of the weakly degenerate eliptic operator associatd with spectral quantity tilde(B), with main characteristics
//fit and fit2 (Suitable for tensorial resolution of BH spacetime)
//See also function tilde(laplacian)
void tensorellipticBt( Scalar source, Scalar& resu, double fit, double fit2, double fit0d2, double fit1d2, double fit0d3, double fit1d3) {
const int nz = (*source.get_mp().get_mg()).get_nzone(); // Number of domains
int nr = (*source.get_mp().get_mg()).get_nr(1); // Number of collocation points in r in each domain
int nt = (*source.get_mp().get_mg()).get_nt(1); // Number of collocation points in theta in each domain
int np = (*source.get_mp().get_mg()).get_np(1); // Number of collocation points in phi in each domain
const Map_af* map = dynamic_cast<const Map_af*>(&source.get_mp()) ;
const Mg3d* mgrid = (*map).get_mg();
// Some helpful stuff...
const Coord& rr = (*map).r ;
Scalar rrr (*map) ;
rrr = rr ;
rrr.set_spectral_va().set_base(source.get_spectral_va().base);
Scalar source_coq = source ;
source_coq.mult_r() ;
source_coq.mult_r() ;
source.set_spectral_va().ylm() ;
source_coq.set_spectral_va().ylm() ;
Scalar phi(source.get_mp()) ;
phi.annule_hard() ;
// phi.std_spectral_base();
phi.set_spectral_va().set_base(source.get_spectral_va().base) ;
phi.set_spectral_va().ylm() ;
Mtbl_cf& sol_coef = (*phi.set_spectral_va().c_cf) ;
const Base_val& base = source.get_spectral_base() ;
Mtbl_cf sol_part(mgrid, base) ; sol_part.annule_hard() ;
Mtbl_cf sol_hom1(mgrid, base) ; sol_hom1.annule_hard() ;
Mtbl_cf sol_hom2(mgrid, base) ; sol_hom2.annule_hard() ;
int l_q, m_q, base_r ;
{ int lz = 0 ;
for (int k=0; k < np; k++)
for (int j=0; j<nt; j++) {
base.give_quant_numbers(lz, k, j, m_q, l_q, base_r) ;
if (nullite_plm(j, nt, k, np, base) == 1) {
for (int ii=0 ; ii<nr ; ii++){
sol_hom1.set(lz, k, j, ii) = 0 ;
sol_part.set(lz, k, j, ii) = 0 ;
}
}
}
}
{ int lz = 1 ; // The first shell is a really particular case, where the operator is different, and homogeneous solutions have to be handled really carefully.
double alpha = (*map).get_alpha()[lz] ;
double beta = (*map).get_beta()[lz] ;
double ech = beta / alpha ;
for (int k=0; k < np; k++)
for (int j=0; j<nt; j++) {
base.give_quant_numbers(lz, k, j, m_q, l_q, base_r) ;
if (nullite_plm(j, nt, k, np, base) == 1) {
Matrice ope(nr,nr) ;
ope.annule_hard() ;
Diff_dsdx dx(base_r, nr) ; const Matrice& mdx = dx.get_matrice() ;
Diff_dsdx2 dx2(base_r, nr) ; const Matrice& mdx2 = dx2.get_matrice() ;
Diff_id id(base_r, nr) ; const Matrice& mid = id.get_matrice() ;
Diff_xdsdx xdx(base_r, nr) ; const Matrice& mxdx = xdx.get_matrice() ;
Diff_xdsdx2 xdx2(base_r, nr) ; const Matrice& mxdx2 = xdx2.get_matrice() ;
Diff_x2dsdx2 x2dx2(base_r, nr) ; const Matrice& mx2dx2 = x2dx2.get_matrice() ;
Diff_x3dsdx2 x3dx2 (base_r, nr); const Matrice& mx3dx2 = x3dx2.get_matrice();
Diff_x4dsdx2 x4dx2 (base_r, nr); const Matrice& mx4dx2 = x4dx2.get_matrice();
// ope = mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2 + 2*(mxdx + ech*mdx)
// - l_q*(l_q+1)*mid - (mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) - (fit)*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) + (fit)*beta*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) + (fit)*alpha*(mx3dx2 + 2*ech*mx2dx2 + ech*ech*mxdx2) ;
// ope = mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2 + 2*(mxdx + ech*mdx)
// - l_q*(l_q+1)*mid - ((mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) -(fit)*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) + (fit)*beta*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2)
// + (fit)*alpha*(mx3dx2 + 2*ech*mx2dx2 + ech*ech*mxdx2) + fit2*alpha*alpha*(mx4dx2 + 2*ech*mx3dx2 + ech*ech*mx2dx2)
// + 2*fit2*alpha*(beta-1.)*(mx3dx2 + 2*ech*mx2dx2 + ech*ech*mxdx2) + fit2*(beta-1.)*(beta-1.)*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2));
ope = mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2 + 2*(mxdx + ech*mdx)
- l_q*(l_q+1)*mid + 2*l_q*mid - ((mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) -(fit)*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) + (fit)*beta*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2)
+ (fit)*alpha*(mx3dx2 + 2*ech*mx2dx2 + ech*ech*mxdx2) + fit2*alpha*alpha*(mx4dx2 + 2*ech*mx3dx2 + ech*ech*mx2dx2)
+ fit2*alpha*(beta-1.)*(mx3dx2 + 2*ech*mx2dx2 + ech*ech*mxdx2) + fit2*alpha*(beta- rrr.val_grid_point(1,0,0, nr-1))*(mx3dx2 + 2*ech*mx2dx2 + ech*ech*mxdx2)+ fit2*(beta-1.)*(beta- rrr.val_grid_point(1,0,0,nr -1))*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2));
for (int col=0; col<nr; col++)
ope.set(nr-1, col) = 0 ;
ope.set(nr-1, 0) = 1 ;
Tbl rhs(nr);
rhs.annule_hard() ;
for (int i=0; i<nr; i++)
rhs.set(i) = (*source_coq.get_spectral_va().c_cf)(1, k, j, i) ;
rhs.set(nr-1) = 0 ;
ope.set_lu() ;
Tbl sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_part.set(1, k, j, i) = sol(i) ;
rhs.annule_hard();
rhs.set(nr-1) = 1 ;
sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_hom1.set(1, k, j, i) = sol(i) ;
}
}
}
// Attention! zones 2 et 3traitee separement egalement!!!
// Current implementations only allow grids with more than 3 shells.
{ int lz = 2 ;
double alpha = (*map).get_alpha()[lz] ;
double beta = (*map).get_beta()[lz] ;
double ech = beta / alpha ;
for (int k=0; k < np; k++)
for (int j=0; j<nt; j++) {
base.give_quant_numbers(lz, k, j, m_q, l_q, base_r) ;
if (nullite_plm(j, nt, k, np, base) == 1) {
Matrice ope(nr,nr) ;
ope.annule_hard() ;
Diff_dsdx dx(base_r, nr) ; const Matrice& mdx = dx.get_matrice() ;
Diff_dsdx2 dx2(base_r, nr) ; const Matrice& mdx2 = dx2.get_matrice() ;
Diff_id id(base_r, nr) ; const Matrice& mid = id.get_matrice() ;
Diff_xdsdx xdx(base_r, nr) ; const Matrice& mxdx = xdx.get_matrice() ;
Diff_xdsdx2 xdx2(base_r, nr) ; const Matrice& mxdx2 = xdx2.get_matrice() ;
Diff_x2dsdx2 x2dx2(base_r, nr) ; const Matrice& mx2dx2 = x2dx2.get_matrice() ;
Diff_x3dsdx2 x3dx2 (base_r, nr); const Matrice& mx3dx2 = x3dx2.get_matrice();
ope = mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2 + 2*(mxdx + ech*mdx)
- l_q*(l_q+1)*mid + 2*l_q*mid - fit0d2*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) + (fit1d2)*(rrr.val_grid_point(lz, 0, 0, 0))*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) - (fit1d2)*beta*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) - (fit1d2)*alpha*(mx3dx2 + 2*ech*mx2dx2 + ech*ech*mxdx2) ;
// ope = mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2 + 2*(mxdx + ech*mdx)
// - l_q*(l_q+1)*mid ;
for (int col=0; col<nr; col++)
ope.set(nr-1, col) = 0 ;
ope.set(nr-1, 0) = 1 ;
for (int col=0; col<nr; col++) {
ope.set(nr-2, col) = 0 ;
}
ope.set(nr-2, 1) = 1 ;
Tbl rhs(nr) ;
rhs.annule_hard() ;
for (int i=0; i<nr; i++)
rhs.set(i) = (*source_coq.get_spectral_va().c_cf)(lz, k, j, i) ;
rhs.set(nr-2) = 0 ;
rhs.set(nr-1) = 0 ;
ope.set_lu() ;
Tbl sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_part.set(lz, k, j, i) = sol(i) ;
rhs.annule_hard() ;
rhs.set(nr-2) = 1 ;
sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_hom1.set(lz, k, j, i) = sol(i) ;
rhs.set(nr-2) = 0 ;
rhs.set(nr-1) = 1 ;
sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_hom2.set(lz, k, j, i) = sol(i) ;
}
}
}
{ int lz = 3 ;
double alpha = (*map).get_alpha()[lz] ;
double beta = (*map).get_beta()[lz] ;
double ech = beta / alpha ;
for (int k=0; k < np; k++)
for (int j=0; j<nt; j++) {
base.give_quant_numbers(lz, k, j, m_q, l_q, base_r) ;
if (nullite_plm(j, nt, k, np, base) == 1) {
Matrice ope(nr,nr) ;
ope.annule_hard() ;
Diff_dsdx dx(base_r, nr) ; const Matrice& mdx = dx.get_matrice() ;
Diff_dsdx2 dx2(base_r, nr) ; const Matrice& mdx2 = dx2.get_matrice() ;
Diff_id id(base_r, nr) ; const Matrice& mid = id.get_matrice() ;
Diff_xdsdx xdx(base_r, nr) ; const Matrice& mxdx = xdx.get_matrice() ;
Diff_xdsdx2 xdx2(base_r, nr) ; const Matrice& mxdx2 = xdx2.get_matrice() ;
Diff_x2dsdx2 x2dx2(base_r, nr) ; const Matrice& mx2dx2 = x2dx2.get_matrice() ;
Diff_x3dsdx2 x3dx2 (base_r, nr); const Matrice& mx3dx2 = x3dx2.get_matrice();
ope = mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2 + 2*(mxdx + ech*mdx)
- l_q*(l_q+1)*mid +2*l_q*mid - fit0d3*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) + (fit1d3)*(rrr.val_grid_point(lz, 0, 0, 0))*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) - (fit1d3)*beta*(mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2) - (fit1d3)*alpha*(mx3dx2 + 2*ech*mx2dx2 + ech*ech*mxdx2) ;
// ope = mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2 + 2*(mxdx + ech*mdx)
// - l_q*(l_q+1)*mid ;
for (int col=0; col<nr; col++)
ope.set(nr-1, col) = 0 ;
ope.set(nr-1, 0) = 1 ;
for (int col=0; col<nr; col++) {
ope.set(nr-2, col) = 0 ;
}
ope.set(nr-2, 1) = 1 ;
Tbl rhs(nr) ;
rhs.annule_hard() ;
for (int i=0; i<nr; i++)
rhs.set(i) = (*source_coq.get_spectral_va().c_cf)(lz, k, j, i) ;
rhs.set(nr-2) = 0 ;
rhs.set(nr-1) = 0 ;
ope.set_lu() ;
Tbl sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_part.set(lz, k, j, i) = sol(i) ;
rhs.annule_hard() ;
rhs.set(nr-2) = 1 ;
sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_hom1.set(lz, k, j, i) = sol(i) ;
rhs.set(nr-2) = 0 ;
rhs.set(nr-1) = 1 ;
sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_hom2.set(lz, k, j, i) = sol(i) ;
}
}
}
// Current implementations only allow grids with more than 2 shells.
for (int lz=4; lz<nz-1; lz++) {
double ech = (*map).get_beta()[lz] / (*map).get_alpha()[lz] ;
for (int k=0; k < np; k++)
for (int j=0; j<nt; j++) {
base.give_quant_numbers(lz, k, j, m_q, l_q, base_r) ;
if (nullite_plm(j, nt, k, np, base) == 1) {
Matrice ope(nr,nr) ;
ope.annule_hard() ;
Diff_dsdx dx(base_r, nr) ; const Matrice& mdx = dx.get_matrice() ;
Diff_dsdx2 dx2(base_r, nr) ; const Matrice& mdx2 = dx2.get_matrice() ;
Diff_id id(base_r, nr) ; const Matrice& mid = id.get_matrice() ;
Diff_xdsdx xdx(base_r, nr) ; const Matrice& mxdx = xdx.get_matrice() ;
Diff_xdsdx2 xdx2(base_r, nr) ; const Matrice& mxdx2 = xdx2.get_matrice() ;
Diff_x2dsdx2 x2dx2(base_r, nr) ; const Matrice& mx2dx2 = x2dx2.get_matrice() ;
ope = mx2dx2 + 2*ech*mxdx2 + ech*ech*mdx2 + 2*(mxdx + ech*mdx)
- l_q*(l_q+1)*mid +2*l_q*mid ;
for (int col=0; col<nr; col++)
ope.set(nr-1, col) = 0 ;
ope.set(nr-1, 0) = 1 ;
for (int col=0; col<nr; col++) {
ope.set(nr-2, col) = 0 ;
}
ope.set(nr-2, 1) = 1 ;
Tbl rhs(nr) ;
rhs.annule_hard() ;
for (int i=0; i<nr; i++)
rhs.set(i) = (*source_coq.get_spectral_va().c_cf)(lz, k, j, i) ;
rhs.set(nr-2) = 0 ;
rhs.set(nr-1) = 0 ;
ope.set_lu() ;
Tbl sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_part.set(lz, k, j, i) = sol(i) ;
rhs.annule_hard() ;
rhs.set(nr-2) = 1 ;
sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_hom1.set(lz, k, j, i) = sol(i) ;
rhs.set(nr-2) = 0 ;
rhs.set(nr-1) = 1 ;
sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_hom2.set(lz, k, j, i) = sol(i) ;
}
}
}
{ int lz = nz-1 ;
double alpha = (*map).get_alpha()[lz] ;
for (int k=0; k < np; k++)
for (int j=0; j<nt; j++) {
base.give_quant_numbers(lz, k, j, m_q, l_q, base_r) ;
if (nullite_plm(j, nt, k, np, base) == 1) {
Matrice ope(nr,nr) ;
ope.annule_hard() ;
Diff_dsdx2 dx2(base_r, nr) ; const Matrice& mdx2 = dx2.get_matrice() ;
Diff_sx2 sx2(base_r, nr) ; const Matrice& ms2 = sx2.get_matrice() ;
ope = (mdx2 - l_q*(l_q+1)*ms2 + 2*l_q*ms2)/(alpha*alpha) ;
for (int i=0; i<nr; i++)
ope.set(nr-1, i) = 0 ;
ope.set(nr-1, 0) = 1 ; //for the true homogeneous solution
for (int i=0; i<nr; i++) {
ope.set(nr-2, i) = 1 ; //for the limit at inifinity
}
if (l_q > 0) {
for (int i=0; i<nr; i++) {
ope.set(nr-3, i) = i*i ; //for the finite part (derivative = 0 at infty)
}
}
// cout << "l: " << l_q << endl ;
Tbl rhs(nr) ;
rhs.annule_hard() ;
for (int i=0; i<nr; i++)
rhs.set(i) = (*source.get_spectral_va().c_cf)(lz, k, j, i) ;
if (l_q>0) rhs.set(nr-3) = 0 ;
rhs.set(nr-2) = 0 ;
rhs.set(nr-1) = 0 ;
ope.set_lu() ;
Tbl sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_part.set(lz, k, j, i) = sol(i) ;
rhs.annule_hard() ;
rhs.set(nr-1) = 1 ;
sol = ope.inverse(rhs) ;
for (int i=0; i<nr; i++)
sol_hom2.set(lz, k, j, i) = sol(i) ;
}
}
}
Mtbl_cf dpart = sol_part ; dpart.dsdx() ;
Mtbl_cf dhom1 = sol_hom1 ; dhom1.dsdx() ;
Mtbl_cf dhom2 = sol_hom2 ; dhom2.dsdx() ;
// Now matching the homogeneous solutions between the different domains...
for (int k=0; k < np; k++)
for (int j=0; j<nt; j++) {
base.give_quant_numbers(0, k, j, m_q, l_q, base_r) ;
if (nullite_plm(j, nt, k, np, base) == 1) {
Matrice systeme(2*nz-4, 2*nz-4) ;
systeme.annule_hard() ;
Tbl rhs(2*nz-4) ;
rhs.annule_hard() ;
//First shell
int lin = 0 ;
int col = 0 ;
double alpha = (*map).get_alpha()[1] ;
systeme.set(lin, col) += sol_hom1.val_out_bound_jk(1, j, k) ;
rhs.set(lin) -= sol_part.val_out_bound_jk(1, j, k) ;
lin++ ;
systeme.set(lin, col) += dhom1.val_out_bound_jk(1, j, k) / alpha ;
rhs.set(lin) -= dpart.val_out_bound_jk(1, j, k) / alpha ;
col += 1 ;
//Shells
for (int lz=2; lz<nz-1; lz++) {
alpha = (*map).get_alpha()[lz] ;
lin-- ;
systeme.set(lin,col) -= sol_hom1.val_in_bound_jk(lz, j, k) ;
systeme.set(lin,col+1) -= sol_hom2.val_in_bound_jk(lz, j, k) ;
rhs.set(lin) += sol_part.val_in_bound_jk(lz, j, k) ;
lin++ ;
systeme.set(lin,col) -= dhom1.val_in_bound_jk(lz, j, k) / alpha ;
systeme.set(lin,col+1) -= dhom2.val_in_bound_jk(lz, j, k) / alpha ;
rhs.set(lin) += dpart.val_in_bound_jk(lz, j, k) / alpha;
lin++ ;
systeme.set(lin, col) += sol_hom1.val_out_bound_jk(lz, j, k) ;
systeme.set(lin, col+1) += sol_hom2.val_out_bound_jk(lz, j, k) ;
rhs.set(lin) -= sol_part.val_out_bound_jk(lz, j, k) ;
lin++ ;
systeme.set(lin, col) += dhom1.val_out_bound_jk(lz, j, k) / alpha ;
systeme.set(lin, col+1) += dhom2.val_out_bound_jk(lz, j, k) / alpha ;
rhs.set(lin) -= dpart.val_out_bound_jk(lz, j, k) / alpha ;
col += 2 ;
}
//CED
alpha = (*map).get_alpha()[nz-1] ;
lin-- ;
systeme.set(lin,col) -= sol_hom2.val_in_bound_jk(nz-1, j, k) ;
rhs.set(lin) += sol_part.val_in_bound_jk(nz-1, j, k) ;
lin++ ;
systeme.set(lin,col) -= (-4*alpha)*dhom2.val_in_bound_jk(nz-1, j, k) ;
rhs.set(lin) += (-4*alpha)*dpart.val_in_bound_jk(nz-1, j, k) ;
systeme.set_lu() ;
// cout << systeme << endl;
Tbl coef = systeme.inverse(rhs);
int indice = 0 ;
// int tryr; cin >> tryr;
// for (int i=0; i<mgrid.get_nr(0); i++)
// sol_coef.set(0, k, j, i) = 0 ;
// sol_coef.set(0, k, j, 0) = (*bound.get_spectral_va().c_cf)(0, k, j, 0) ;
for (int i=0; i<(*mgrid).get_nr(1); i++)
sol_coef.set(1, k, j, i) = sol_part(1, k, j, i)
+coef(indice)*sol_hom1(1, k, j, i) ;
indice +=1;
for (int lz=2; lz<nz-1; lz++) {
for (int i=0; i<(*mgrid).get_nr(lz); i++)
sol_coef.set(lz, k, j, i) = sol_part(lz, k, j, i)
+coef(indice)*sol_hom1(lz, k, j, i)
+coef(indice+1)*sol_hom2(lz, k, j, i) ;
indice += 2 ;
}
for (int i=0; i<(*mgrid).get_nr(nz-1); i++)
sol_coef.set(nz-1, k, j, i) = sol_part(nz-1, k, j, i)
+coef(indice)*sol_hom2(nz-1, k, j, i) ;
}
}
delete phi.set_spectral_va().c ;
phi.set_spectral_va().c = 0x0 ;
// phi.set_spectral_va().ylm_i() ;
phi.annule_domain(nz-1);
resu = phi;
}
}
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