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#include "cctk.h"
#include "cctk_Arguments.h"
#include "cctk_Functions.h"
#include "cctk_Parameters.h"
subroutine qlm_calc_coordinates (CCTK_ARGUMENTS, hn)
use cctk
use constants
use qlm_boundary
use tensor2
implicit none
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_FUNCTIONS
DECLARE_CCTK_PARAMETERS
integer :: hn
CCTK_REAL, parameter :: one=1, two=2, half=one/two
CCTK_REAL :: z0, z0dot, z1, z1dot
CCTK_REAL :: qq(2,2), dtq
CCTK_REAL :: integral_z, area, radius
integer :: i0,j0
integer :: i,j
if (veryverbose/=0) then
call CCTK_INFO ("Finding invariant coordinates")
end if
! latitude of "equator"
i0 = (qlm_ntheta(hn)+1)/2
! longitude of zero meridian
j0 = 1+qlm_nghostsphi(hn)
! calculate area
area = 0
do j = 1+qlm_nghostsphi(hn), qlm_nphi(hn)-qlm_nghostsphi(hn)
do i = 1+qlm_nghoststheta(hn), qlm_ntheta(hn)-qlm_nghoststheta(hn)
! 2-metric on the horizon
qq(1,1) = qlm_qtt(i,j,hn)
qq(1,2) = qlm_qtp(i,j,hn)
qq(2,2) = qlm_qpp(i,j,hn)
qq(2,1) = qq(1,2)
call calc_2det (qq, dtq)
area = area + sqrt(dtq) * qlm_delta_theta(hn) * qlm_delta_phi(hn)
end do
end do
radius = sqrt(area / (4*pi))
! initial value
qlm_inv_z(i0,j0,hn) = 0
! d_a z = 1/R^2 xi^b eps_ba
! transport along equator
do j = j0+1, qlm_nphi(hn)-qlm_nghostsphi(hn)
i = i0
z0 = qlm_inv_z(i,j-1,hn)
z0dot = rhs(i,j-1,(/0,1/))
z1 = z0 + qlm_delta_phi(hn) * z0dot
z1dot = rhs(i,j,(/0,1/))
qlm_inv_z(i,j,hn) = z0 + half * qlm_delta_phi(hn) * (z0dot + z1dot)
end do
! transport along meridians
do j = 1+qlm_nghostsphi(hn), qlm_nphi(hn)-qlm_nghostsphi(hn)
do i = i0-1, 1+qlm_nghoststheta(hn), -1
z0 = qlm_inv_z(i+1,j,hn)
z0dot = rhs(i+1,j,(/-1,0/))
z1 = z0 + qlm_delta_theta(hn) * z0dot
z1dot = rhs(i,j,(/-1,0/))
qlm_inv_z(i,j,hn) = z0 + half * qlm_delta_theta(hn) * (z0dot + z1dot)
end do
do i = i0+1, qlm_ntheta(hn)-qlm_nghoststheta(hn)
z0 = qlm_inv_z(i-1,j,hn)
z0dot = rhs(i-1,j,(/1,0/))
z1 = z0 + qlm_delta_theta(hn) * z0dot
z1dot = rhs(i,j,(/1,0/))
qlm_inv_z(i,j,hn) = z0 + half * qlm_delta_theta(hn) * (z0dot + z1dot)
end do
end do
! normalise
integral_z = 0
do j = 1+qlm_nghostsphi(hn), qlm_nphi(hn)-qlm_nghostsphi(hn)
do i = 1+qlm_nghoststheta(hn), qlm_ntheta(hn)-qlm_nghoststheta(hn)
! 2-metric on the horizon
qq(1,1) = qlm_qtt(i,j,hn)
qq(1,2) = qlm_qtp(i,j,hn)
qq(2,2) = qlm_qpp(i,j,hn)
qq(2,1) = qq(1,2)
call calc_2det (qq, dtq)
integral_z = integral_z + qlm_inv_z(i,j,hn) * sqrt(dtq) * qlm_delta_theta(hn) * qlm_delta_phi(hn)
end do
end do
qlm_inv_z(:,:,hn) = qlm_inv_z(:,:,hn) - integral_z / area
! boundary conditions
call set_boundary (CCTK_PASS_FTOF, hn, qlm_inv_z(:,:,hn), +1)
#if 0
! initial value
qlm_inv_phi(i0,j0,hn) = 0
! xi^a d_a phi = C
! z^a d_a phi = 0
! z^a = (R^4 / q_bc xi^b xi^c) q^ab d_b z
! v^a = A z^a + B xi^a
! v^a d_a phi = (A z^a + B xi^a) d_a phi
! = B C
! (choose C=1, and normalise later)
#error "replace z by phi"
! transport along equator
do j = j0+1, qlm_nphi(hn)-qlm_nghostsphi(hn)
i = i0
z0 = qlm_inv_z(i,j-1,hn)
z0dot = rhs(i,j-1,(/0,1/))
z1 = z0 + qlm_delta_phi(hn) * z0dot
z1dot = rhs(i,j,(/0,1/))
qlm_inv_z(i,j,hn) = z0 + half * qlm_delta_phi(hn) * (z0dot + z1dot)
end do
! transport along meridians
do j = 1+qlm_nghostsphi(hn), qlm_nphi(hn)-qlm_nghostsphi(hn)
do i = i0-1, 1+qlm_nghoststheta(hn), -1
z0 = qlm_inv_z(i+1,j,hn)
z0dot = rhs(i+1,j,(/-1,0/))
z1 = z0 + qlm_delta_theta(hn) * z0dot
z1dot = rhs(i,j,(/-1,0/))
qlm_inv_z(i,j,hn) = z0 + half * qlm_delta_theta(hn) * (z0dot + z1dot)
end do
do i = i0+1, qlm_ntheta(hn)-qlm_nghoststheta(hn)
z0 = qlm_inv_z(i-1,j,hn)
z0dot = rhs(i-1,j,(/1,0/))
z1 = z0 + qlm_delta_theta(hn) * z0dot
z1dot = rhs(i,j,(/1,0/))
qlm_inv_z(i,j,hn) = z0 + half * qlm_delta_theta(hn) * (z0dot + z1dot)
end do
end do
#error "normalise"
#endif
contains
function rhs (i, j, vv) result (zdot)
integer, intent(in) :: i, j
integer, intent(in) :: vv(2)
CCTK_REAL :: zdot
CCTK_REAL :: qq(2,2), dtq
! 2-metric on the horizon
qq(1,1) = qlm_qtt(i,j,hn)
qq(1,2) = qlm_qtp(i,j,hn)
qq(2,2) = qlm_qpp(i,j,hn)
qq(2,1) = qq(1,2)
call calc_2det (qq, dtq)
zdot = vv(1) * (- (1/radius**2) * qlm_xi_p(i,j,hn) * sqrt(dtq)) &
+ vv(2) * (+ (1/radius**2) * qlm_xi_t(i,j,hn) * sqrt(dtq))
end function rhs
end subroutine qlm_calc_coordinates
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