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#include "cctk.h"
#include "cctk_Arguments.h"
#include "cctk_Functions.h"
#include "cctk_Parameters.h"
subroutine qlm_killing_gradient (CCTK_ARGUMENTS, hn)
use cctk
use constants
use qlm_boundary
use qlm_derivs
use tensor2
implicit none
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_FUNCTIONS
DECLARE_CCTK_PARAMETERS
integer :: hn
CCTK_REAL, parameter :: two=2, half=1/two
CCTK_REAL :: qq(2,2), dqq(2,2,2), dtq, qu(2,2), dqu(2,2,2)
CCTK_REAL :: dpsi2(2), ddpsi2(2,2), ndpsi2, dndpsi2(2)
CCTK_REAL :: xi(2), dxi(2,2), chi
integer :: i, j
integer :: a, b
CCTK_REAL :: delta_space(2)
delta_space(:) = (/ qlm_delta_theta(hn), qlm_delta_phi(hn) /)
! Calculate the gradient of a scalar
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)
#if 0
dqq(1,1,1) = qlm_dqttt(i,j,hn)
dqq(1,1,2) = qlm_dqttp(i,j,hn)
dqq(1,2,1) = qlm_dqtpt(i,j,hn)
dqq(1,2,2) = qlm_dqtpp(i,j,hn)
dqq(2,2,1) = qlm_dqppt(i,j,hn)
dqq(2,2,2) = qlm_dqppp(i,j,hn)
dqq(2,1,:) = dqq(1,2,:)
#endif
call calc_2det (qq, dtq)
#if 0
call calc_2inv (qq, dtq, qu)
call calc_2invderiv (qu, dqq, dqu)
#endif
dpsi2(1) = (abs2(qlm_psi2(i+1,j,hn)) - abs2(qlm_psi2(i-1,j,hn))) / (2*qlm_delta_theta(hn))
dpsi2(2) = (abs2(qlm_psi2(i,j+1,hn)) - abs2(qlm_psi2(i,j-1,hn))) / (2*qlm_delta_phi(hn))
#if 0
ddpsi2(1,1) = (abs2(qlm_psi2(i+1,j,hn)) - 2*abs2(qlm_psi2(i,j,hn)) + abs2(qlm_psi2(i-1,j,hn))) / qlm_delta_theta(hn)**2
ddpsi2(2,2) = (abs2(qlm_psi2(i,j+1,hn)) - 2*abs2(qlm_psi2(i,j,hn)) + abs2(qlm_psi2(i,j-1,hn))) / qlm_delta_phi(hn)**2
ddpsi2(1,1) = (abs2(qlm_psi2(i-1,j-1,hn)) - abs2(qlm_psi2(i+1,j-1,hn)) - abs2(qlm_psi2(i-1,j+1,hn)) + abs2(qlm_psi2(i+1,j+1,hn))) / (4*qlm_delta_theta(hn)*qlm_delta_phi(hn))
ddpsi2(2,1) = ddpsi2(1,2)
! ndpsi2 = ||grad |Psi_2|^2||
ndpsi2 = 0
do a=1,2
do b=1,2
ndpsi2 = ndpsi2 + qu(a,b) * dpsi2(a) * dpsi2(b)
end do
end do
ndpsi2 = sqrt(ndpsi2)
! dndpsi2 = grad ||grad |Psi_2|^2||
do a=1,2
dndpsi2(a) = 0
do b=1,2
do c=1,2
dndpsi2(a) = dndpsi2(a) + 1 / (2*ndpsi2) * (qu(b,c) * ddpsi2(b,a) * dpsi2(c) + qu(b,c) * dpsi2(b) * ddpsi2(c,a) + dqu(b,c,a) * dpsi2(b) * dpsi2(c))
end do
end do
end do
#endif
! xi^a = eps^ab D_b |Psi_2|^2
do a=1,2
xi(a) = 0
do b=1,2
xi(a) = xi(a) + sqrt(dtq) * epsilon2(a,b) * dpsi2(b)
end do
end do
qlm_xi_t(i,j,hn) = xi(1)
qlm_xi_p(i,j,hn) = xi(2)
#if 0
! xi^a = eps^ab D_b ||D_c |Psi_2|^2||
do a=1,2
xi(a) = 0
do b=1,2
xi(a) = xi(a) + sqrt(dtq) * epsilon2(a,b) * dndpsi2(b)
end do
end do
qlm_xi_t(i,j,hn) = xi(1)
qlm_xi_p(i,j,hn) = xi(2)
#endif
end do
end do
call set_boundary (CCTK_PASS_FTOF, hn, qlm_xi_t(:,:,hn), -1)
call set_boundary (CCTK_PASS_FTOF, hn, qlm_xi_p(:,:,hn), -1)
! fix up xi (which must not be zero)
do j = 1+qlm_nghostsphi(hn), qlm_nphi(hn)-qlm_nghostsphi(hn)
do i = 1+qlm_nghoststheta(hn), qlm_ntheta(hn)-qlm_nghoststheta(hn)
xi(1) = qlm_xi_t(i,j,hn)
xi(2) = qlm_xi_p(i,j,hn)
if (sum(xi**2) < 1.0d-4**2) then
qlm_xi_t(i,j,hn) = sum(qlm_xi_t(i:i+1,j:j+1,hn)) / 4
qlm_xi_p(i,j,hn) = sum(qlm_xi_p(i:i+1,j:j+1,hn)) / 4
end if
end do
end do
call set_boundary (CCTK_PASS_FTOF, hn, qlm_xi_t(:,:,hn), -1)
call set_boundary (CCTK_PASS_FTOF, hn, qlm_xi_p(:,:,hn), -1)
! set up the derivative of xi (which is not really needed)
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)
dxi(1,1:2) = deriv (qlm_xi_t(:,:,hn), i, j, delta_space)
dxi(2,1:2) = deriv (qlm_xi_p(:,:,hn), i, j, delta_space)
! eps_ab sqrt(q) chi = D_b xi_a
! sqrt(q) chi = -1/2 eps^ab D_a xi_b
chi = 0
do a=1,2
do b=1,2
chi = chi - half * sqrt(dtq) * epsilon2(a,b) * dxi(b,a)
end do
end do
qlm_chi(i,j,hn) = chi
end do
end do
call set_boundary (CCTK_PASS_FTOF, hn, qlm_chi (:,:,hn), +1)
end subroutine qlm_killing_gradient
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