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/*@@
@file AHFinder_fun.F
@date April 1998
@author Miguel Alcubierre
@desc
Find horizon function.
@enddesc
@@*/
#include "cctk.h"
#include "cctk_Parameters.h"
#include "cctk_Arguments.h"
subroutine AHFinder_fun(CCTK_ARGUMENTS)
use AHFinder_dat
implicit none
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_PARAMETERS
DECLARE_CCTK_FUNCTIONS
integer i,j,k
integer l,m,ll
CCTK_REAL LEGEN
CCTK_REAL xp,yp,zp,rp
CCTK_REAL phi,cost,cosa,sina
CCTK_REAL zero,half,one,two
CCTK_REAL pi,halfpi,twopi
CCTK_REAL aux1,aux2
CCTK_REAL red_tmp
CCTK_REAL local_legen,local_legen_old,local_legen_old_old
CCTK_REAL factor
CCTK_REAL cc_cosa(lmax,lmax),cs_sina(lmax,lmax)
CCTK_REAL dsqrt_factor(lmax,lmax),dsqrt_factor2(lmax)
CCTK_REAL lm_factor1(0:lmax,0:lmax),lm_factor2(0:lmax,0:lmax)
CCTK_REAL lm_factor3(lmax)
CCTK_REAL sinphi,cosphi,sinphi1,cosphi1,sinphi2,cosphi2
CCTK_REAL legen_base
! **************************
! *** DEFINE NUMBERS ***
! **************************
zero = 0.0D0
half = 0.5D0
one = 1.0D0
two = 2.0D0
pi = acos(-one)
halfpi = half*pi
twopi = two*pi
! *******************************************
! *** FACTORS FOR SPHERICAL HARMONICS ***
! *******************************************
! Compute the normalization factors only once for all (l,m)
!
! For M=0 I use the normalization factor:
!
! sqrt(2*L + 1)
!
! For M non-zero I use the normalization factor:
!
! sqrt( 2 (2*L + 1) (L-M)! / (L+M)! )
!
! The extra factor of sqrt(2) is there because I
! use a basis of sines and cosines and not the
! standard complex exponential.
!
! With this normalization, I ensure that my basis
! functions f _lm are such that:
!
! /
! | f_lm f_l;mp sin(theta) dtheta dphi = 4 pi delta_mm' delta_ll'
! /
!
! Notice the extra factor of 4 pi. This is there because
! for a sphere, I want the (0,0) coefficient to correspond
! to the radius.
!
! Also compute the multiplying factors in the recursion relations
! Initialize.
lm_factor1 = zero
lm_factor2 = zero
lm_factor3 = zero
dsqrt_factor = zero
dsqrt_factor2 = zero
! M=0.
do l=1,lmax
dsqrt_factor2(l) = c0(l)*sqrt(dble(2*l+1))
lm_factor1(l,0) = dble(2*l-1)/dble(l)
lm_factor2(l,0) = dble(l-1)/dble(l)
lm_factor3(l) = dble(2*l-1)
end do
! M > 0.
if (nonaxi) then
do l=1,lmax
dsqrt_factor(l,1) = sqrt(2.0D0*dble(2*l+1)/dble(l*(l+1)))
do m=2,l
dsqrt_factor(l,m) = dsqrt_factor(l,m-1)
. / sqrt(dble((l-m+1)*(l+m)))
end do
end do
do l=2,lmax
lm_factor1(l,1) = dble(2*l-1)/dble(l-1)
lm_factor2(l,1) = dble(l)/dble(l-1)
do m=2,l-1
lm_factor1(l,m) = dble(2*l-1)/dble(l-m)
lm_factor2(l,m) = dble(l+m-1)/dble(l-m)
end do
end do
end if
! *********************************
! *** FIND HORIZON FUNCTION ***
! *********************************
! Loop over j.
do j=1,ny
! Find yp.
yp = y(1,j,1) - yc
! Loop over i.
do i=1,nx
! Find xp.
xp = x(i,1,1) - xc
! Find sines and cosines of phi.
if (nonaxi) then
! Find phi.
if (yp.ne.zero) then
phi = atan2(yp,xp)
else
phi = zero
end if
! Compute the expensive sines and cosines only once
! for all (l,m) using the recursion relations:
!
! cos(m*phi) = 2*cos((m-1)*phi)*cos(phi)-cos((m-2)*phi)
! sin(m*phi) = 2*sin((m-1)*phi)*cos(phi)-sin((m-2)*phi)
cosphi = cos(phi)
cosphi1 = one
cosphi2 = cosphi
if (.not.refy) then
sinphi = sin(phi)
sinphi1 = zero
sinphi2 = sinphi
end if
! M=1.
do l=1,lmax
cc_cosa(l,1) = cc(l,1)*cosphi*dsqrt_factor(l,1)
if (.not.refy) then
cs_sina(l,1) = cs(l,1)*sinphi*dsqrt_factor(l,1)
end if
end do
! M>1.
do m=2,lmax
cosa = 2.0D0*cosphi2*cosphi - cosphi1
cosphi1 = cosphi2
cosphi2 = cosa
do l=m,lmax
cc_cosa(l,m) = cc(l,m)*cosa*dsqrt_factor(l,m)
end do
if (.not.refy) then
sina = 2.0D0*sinphi2*cosphi - sinphi1
sinphi1 = sinphi2
sinphi2 = sina
do l=m,lmax
cs_sina(l,m) = cs(l,m)*sina*dsqrt_factor(l,m)
end do
end if
end do
end if
! Loop over k.
do k=1,nz
! Find zp.
zp = z(i,j,k) - zc
! Find rp.
rp = sqrt(xp**2 + yp**2 + zp**2)
! Monopole term.
aux1 = c0(0)
! Axisymmetric terms.
if (rp.ne.zero) then
cost = zp/rp
else
cost = one
end if
! Compute the contribution from M=0, L=1,Lmax using
! the recursion relations from Numerical Recipes.
if (lmax.gt.0) then
! L=1
legen_base = one
local_legen_old_old = one
local_legen_old = cost
aux1 = aux1 + dsqrt_factor2(1)*local_legen_old
! L=2,lmax
do l=2,lmax
local_legen = cost*lm_factor1(l,0)*local_legen_old
. - lm_factor2(l,0)*local_legen_old_old
local_legen_old_old = local_legen_old
local_legen_old = local_legen
aux1 = aux1 + dsqrt_factor2(l)*local_legen
end do
!
! Non-axisymmetric terms.
!
if (nonaxi) then
aux2 = sqrt((one-cost)*(one+cost))
! This general loop can only be used until l=lmax-2 since
! the recursion relations requires 2 starting values.
do m=1,lmax-2
! L=M
legen_base = -legen_base*aux2*lm_factor3(m)
aux1 = aux1 + legen_base*cc_cosa(m,m)
if (.not.refy) then
aux1 = aux1 + legen_base*cs_sina(m,m)
end if
local_legen_old_old = legen_base
! L=M+1
local_legen_old = cost*lm_factor3(m+1)
. *legen_base
aux1 = aux1 + local_legen_old*cc_cosa(m+1,m)
if (.not.refy) then
aux1 = aux1 + local_legen_old*cs_sina(m+1,m)
end if
! L=M+2,lmax
do l=m+2,lmax
local_legen = cost*lm_factor1(l,m)*local_legen_old
. - lm_factor2(l,m)*local_legen_old_old
local_legen_old_old = local_legen_old
local_legen_old = local_legen
aux1 = aux1 + local_legen*cc_cosa(l,m)
if (.not.refy) then
aux1 = aux1 + local_legen*cs_sina(l,m)
end if
end do
end do
! Then do M=lmax-1
if (lmax.gt.1) then
! L=lmax-1
m=lmax-1
legen_base = -legen_base*aux2*lm_factor3(m)
aux1 = aux1 + legen_base*cc_cosa(m,m)
if (.not.refy) then
aux1 = aux1 + legen_base*cs_sina(m,m)
end if
! L=lmax
local_legen = cost*lm_factor3(lmax)*
. legen_base
aux1 = aux1 + local_legen*cc_cosa(lmax,m)
if (.not.refy) then
aux1 = aux1 + local_legen*cs_sina(lmax,m)
end if
end if
! And finally do M=lmax.
! L=lmax
m=lmax
legen_base = -legen_base*aux2*lm_factor3(m)
aux1 = aux1 + legen_base*cc_cosa(m,m)
if (.not.refy) then
aux1 = aux1 + legen_base*cs_sina(m,m)
end if
end if
end if
! Find horizon function.
ahfgrid(i,j,k) = rp - aux1
end do
end do
end do
! ***************
! *** END ***
! ***************
end subroutine AHFinder_fun
|
