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/*@@
@file AHFinder_calcsigma.F
@date July 1999
@author Lars Nerger
@desc
Calculate weight sigma for Nakamura flow
@enddesc
@@*/
#include "cctk.h"
#include "cctk_Parameters.h"
#include "cctk_Arguments.h"
subroutine AHFinder_calcsigma(CCTK_ARGUMENTS,xp,yp,zp,gupij,sigma)
use AHFinder_dat
implicit none
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_PARAMETERS
integer nmax
integer l,m
integer indx1,indx2
integer indx,idir,jdir
CCTK_REAL zero, sigma
CCTK_REAL ni(3),si(3),supi(3),pij(3,3)
CCTK_REAL ylmi((lmax+1)**2,3)
CCTK_REAL alm((lmax+1)**2)
CCTK_REAL xp,yp,zp,er,sum
CCTK_REAL rho,costheta,sintheta,sinthetal(-1:lmax),prod
CCTK_REAL zlm0(0:lmax,0:lmax),zlm1(0:lmax,0:lmax)
CCTK_REAL cosphi,sinphi,cosmphi(0:lmax),sinmphi(0:lmax)
CCTK_REAL thetax,thetay,thetaz
CCTK_REAL phix,phiy
CCTK_REAL c1,c2,c3,metnorm
CCTK_REAL fi(3)
CCTK_REAL gupij(3,3)
! Description of variables
!
! sigma Weight of Nakamura Flow
! **************************
! *** DEFINE NUMBERS ***
! **************************
zero = 0.0D0
nmax = (lmax+1)**2
! ******************************************************
! *** Calculate gradient of Y_lm(x,y,z) ***
! *** Adopted from thorn_SpectralAHF by Gundlach ***
! ******************************************************
! Initialize arrays
ylmi = zero
zlm0 = zero
zlm1 = zero
ni = zero
si = zero
supi = zero
pij = zero
alm = zero
sinthetal = zero
cosmphi = zero
sinmphi = zero
! Calculate quantities in spherical coordinates.
er = sqrt(xp**2 + yp**2 + zp**2)
if (er .eq. 0.d0) then
call CCTK_WARN(0,'r=0 in AHFinder_calcsigma')
end if
C ********PATCH******************************************
C Keep away from the z-axis, artificially.
if (xp**2 + yp**2 .lt. 1.d-16) then
C write(6,*) 'patch on axis in ylmder2'
xp = 0.d0
yp = 1.d-8
end if
C Polar angle theta.
rho = sqrt(xp**2 + yp**2)
costheta = zp / er
sintheta = rho / er
C Array sinthetal(l) contains [sin(theta)]^l
sinthetal(-1) = 0.d0
sinthetal(0) = 1.d0
sinthetal(1) = sintheta
do l=2,lmax
sinthetal(l) = sinthetal(l-1) * sinthetal(1)
end do
C zlm0 is Y_lm stripped of exp(i m phi) and multiplied by sqrt(4pi).
C zlm1 and zlm2 are the first and second derivative of zlm0
C with respect to theta. l=0 and m=l are treated separately.
C We only need zlm0,1,2 for positive m.
C Treat l=0 separately:
zlm0(0,0) = 1.d0
zlm1(0,0) = 0.d0
C Other l>0.
prod = 1.d0
do l=1,lmax
C Initialize m = l:
C c1 = sqrt(2l+1) * sqrt(2l!) / 2^l / l!
C Y_ll = c1 * sin(theta)^l
C Y_llp = c1 * l * sin(theta)^(l-1) * costheta
m = l
prod = - prod * sqrt(dble(2*l-1) / dble(2*l))
c1 = sqrt(dble(2*l+1)) * prod
zlm0(l,m) = c1 * sinthetal(l)
zlm1(l,m) = c1 * dble(l) * sinthetal(l-1) * costheta
C Recursion relations for the other m:
C c2 = sqrt[(2l+1)(2l-1)/(l^2-m^2)]
C c3 = sqrt[((l-1)^2-m^2)(2l=1)/(l^2-m^2)/(2l-3)]
C Y_lm = c2 Y_l-1,m cos(theta) + c3 Y_l-2,m
C Y_lmp = c2 Y_l-1,mp cos(theta) + c2 Y_l-1,m + c3 Y_l-2,mp
m = l-1
c2 = sqrt(dble((2*l+1)*(2*l-1))
$ / dble(l**2-m**2))
zlm0(l,m) = c2 * zlm0(l-1,m) * costheta
zlm1(l,m) = c2 * (zlm1(l-1,m) * costheta
$ - zlm0(l-1,m) * sintheta)
do m=0,l-2
c2 = sqrt(dble((2*l+1)*(2*l-1))
$ / dble(l**2-m**2))
c3 =sqrt(dble((2*l+1)*((l-1)**2-m**2))
$ / dble((l**2 - m**2)*(2*l-3)))
zlm0(l,m) = c2 * zlm0(l-1,m) * costheta
$ - c3 * zlm0(l-2,m)
zlm1(l,m) = c2 * (zlm1(l-1,m) * costheta
$ - zlm0(l-1,m) * sintheta)
$ - c3 * zlm1(l-2,m)
end do
end do
if (rho .ne. 0.d0) then
C Azimuth angle phi is defined.
cosphi = xp / rho
sinphi = yp / rho
C First and second derivatives of theta.
thetax = costheta * cosphi / er
thetay = costheta * sinphi / er
thetaz = - sintheta / er
C First and second derivatives of phi, the angle in the xy-plane.
phix = - sinphi / rho
phiy = cosphi / rho
C Put together arrays of basis functions in the xy plane.
C The sqrt(2) factor is needed in going from the basis exp(+/-imphi)
C to the basis cos/sin(mphi).
cosmphi(0) = 1.d0
sinmphi(0) = 1.d0
cosmphi(1) = sqrt(2.d0) * cosphi
sinmphi(1) = sqrt(2.d0) * sinphi
do m=2,lmax
cosmphi(m) = cosmphi(m-1) * cosphi - sinmphi(m-1) * sinphi
sinmphi(m) = cosmphi(m-1) * sinphi + sinmphi(m-1) * cosphi
end do
else
C See above for a patch that avoids ever getting here.
call CCTK_WARN(0,'ylmder2 cannot handle axis')
C For x = y = 0 the following are not defined.
! cosphi = 7.d77
! sinphi = 7.d77
! thetax = 7.d77
! thetay = 7.d77
! phix = 7.d77
! phiy = 7.d77
! cosmphi = 7.d77
! sinmphi = 7.d77
C Only these are defined.
thetaz = 0.d0
cosmphi(0) = 1.d0
sinmphi(0) = 1.d0
end if
C Assemble the Y_lm from zlm0 and the basis of functions of phi.
C Assemble the Y_lm,i and Y_lm,ij in the corresponding manner.
do l=0,lmax
do m=0,l
C These are the values of indx corresponding to (l,m) and (l,-m).
indx1 = l**2 + 1 + l + m
indx2 = l**2 + 1 + l - m
C Ylm,x
ylmi(indx1,1) = thetax * zlm1(l,m) * cosmphi(m)
$ - dble(m) * phix * zlm0(l,m) * sinmphi(m)
ylmi(indx2,1) = thetax * zlm1(l,m) * sinmphi(m)
$ + dble(m) * phix * zlm0(l,m) * cosmphi(m)
C Ylm,y
ylmi(indx1,2) = thetay * zlm1(l,m) * cosmphi(m)
$ - dble(m) * phiy * zlm0(l,m) * sinmphi(m)
ylmi(indx2,2) = thetay * zlm1(l,m) * sinmphi(m)
$ + dble(m) * phiy * zlm0(l,m) * cosmphi(m)
C Ylm,z
ylmi(indx1,3) = thetaz * zlm1(l,m) * cosmphi(m)
ylmi(indx2,3) = thetaz * zlm1(l,m) * sinmphi(m)
end do
end do
! ********************************
! *** First derivatives of r ***
! ********************************
ni(1) = xp / er
ni(2) = yp / er
ni(3) = zp / er
! *************************************************************
! *** compress c0, cc and cs into alm with single index ***
! *************************************************************
do l=0, lmax
alm(l**2+l+1) = c0(l)
do m=1,lmax
alm(l**2+l+1+m) = cc(l,m)
alm(l**2+l+1-m) = cs(l,m)
end do
end do
! *******************************************************************
! *** Assemble derivatives of f. Recall f = r - h(theta,phi): ***
! *******************************************************************
C Derivatives of r...
C r,i = n_i and r,ij = (delta_ij - n_i n_j) / r = n_ij
do idir=1,3
fi(idir) = ni(idir)
end do
C ...and derivatives of -h.
do indx=1,nmax
do idir=1,3
fi(idir) = fi(idir) - alm(indx) * ylmi(indx,idir)
end do
end do
! *************************************************************
! *** Assemble normal vector s on surfaces of constant f ***
! *** with indices down. That is, normal with respect ***
! *** to the physical metric g_ij. ***
! *************************************************************
metnorm = 0.d0
do idir=1,3
do jdir=1,3
metnorm = metnorm + fi(idir) * fi(jdir)
$ * gupij(idir,jdir)
end do
end do
metnorm = sqrt(metnorm)
do idir=1,3
si(idir) = fi(idir) / metnorm
end do
C Now with indices up, raised by g^ij.
do idir=1,3
supi(idir) = 0.d0
do jdir=1,3
supi(idir) = supi(idir)
$ + gupij(idir,jdir) * si(jdir)
end do
end do
! *******************************************************
! *** Assemble projection operator with indices up ***
! *** p^ij = g^ij - s^i s^j ***
! *******************************************************
do idir=1,3
do jdir=1,3
pij(idir,jdir) = gupij(idir,jdir)
$ - supi(idir) * supi(jdir)
end do
end do
! *************************************************
! *** Calculate weight sigma ***
! *** 2 * r^2 / (p^ij (delta_ij - n_i n_j)) ***
! *************************************************
sum = 0.d0
do idir=1,3
sum = sum + pij(idir,idir)
do jdir=1,3
sum = sum
$ - pij(idir,jdir) * ni(idir) * ni(jdir)
end do
end do
sigma = 2.d0 * er**2 / sum
! ***************
! *** END ***
! ***************
end subroutine AHFinder_calcsigma
|
