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/*@@
@file AHFinder_shift.F
@date May 2000
@author Miguel Alcubierre
@desc
Set up horizon (area) locking shift. The area
locking shift is obtained from the following
expression:
a __a
beta = b \/ f
where f is the "horizon function", and where b is given by:
/ ab 2 \
b = alpha | trK - K d f d f / u |
\ a b /
/ __2 __a__b 2 \
/ | \/ f - \/ \/ f d f d f / u |
\ a b /
with "alpha" the lapse function and:
2 mn
u = g d f d f
m n
@enddesc
@@*/
#include "cctk.h"
#include "cctk_Parameters.h"
#include "cctk_Arguments.h"
#include "cctk_DefineThorn.h"
subroutine AHFinder_shift(CCTK_ARGUMENTS)
use AHFinder_dat
implicit none
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_PARAMETERS
DECLARE_CCTK_FUNCTIONS
integer i,j,k
integer order_save
CCTK_REAL det,idet
CCTK_REAL gdxx,gdyy,gdzz,gdxy,gdxz,gdyz
CCTK_REAL guxx,guyy,guzz,guxy,guxz,guyz
CCTK_REAL kdxx,kdyy,kdzz,kdxy,kdxz,kdyz
CCTK_REAL ddxxx,ddxyy,ddxzz,ddxxy,ddxxz,ddxyz
CCTK_REAL ddyxx,ddyyy,ddyzz,ddyxy,ddyxz,ddyyz
CCTK_REAL ddzxx,ddzyy,ddzzz,ddzxy,ddzxz,ddzyz
CCTK_REAL dfx,dfy,dfz,dfux,dfuy,dfuz
CCTK_REAL d2fxx,d2fyy,d2fzz,d2fxy,d2fxz,d2fyz
CCTK_REAL c2fxx,c2fyy,c2fzz,c2fxy,c2fxz,c2fyz
CCTK_REAL idx,idy,idz,idx2,idy2,idz2,idxy,idxz,idyz
CCTK_REAL zero,half,one,two
CCTK_REAL aux,T0,T1,T2,T3,T4,T5
CCTK_REAL, DIMENSION(nx,ny,nz) :: mask,betax_old,betay_old,betaz_old
! *******************
! *** NUMBERS ***
! *******************
zero = 0.0D0
half = 0.5D0
one = 1.0D0
two = 2.0D0
! *********************************
! *** SAY WHAT WE ARE DOING ***
! *********************************
write(*,*)
write(*,*) 'AHFinder: Setting up horizon locking shift.'
write(*,*)
betax_old = betax
betay_old = betay
betaz_old = betaz
! *******************************
! *** SET UP SHIFT VECTOR ***
! *******************************
idx = half/dx
idy = half/dy
idz = half/dz
idxy = idx*idy
idxz = idx*idz
idyz = idy*idz
idx2 = one/dx**2
idy2 = one/dy**2
idz2 = one/dz**2
do k=2,nz-1
do j=2,ny-1
do i=2,nx-1
! Find spatial metric.
gdxx = gxx(i,j,k)
gdyy = gyy(i,j,k)
gdzz = gzz(i,j,k)
gdxy = gxy(i,j,k)
gdxz = gxz(i,j,k)
gdyz = gyz(i,j,k)
! Find extrinsic curvature.
kdxx = kxx(i,j,k)
kdyy = kyy(i,j,k)
kdzz = kzz(i,j,k)
kdxy = kxy(i,j,k)
kdxz = kxz(i,j,k)
kdyz = kyz(i,j,k)
! Find determinant of spatial metric.
det = gdxx*gdyy*gdzz + two*gdxy*gdxz*gdyz
. - gdxx*gdyz**2 - gdyy*gdxz**2 - gdzz*gdxy**2
idet = one/det
! Find inverse spatial metric.
guxx = idet*(gdyy*gdzz - gdyz**2)
guyy = idet*(gdxx*gdzz - gdxz**2)
guzz = idet*(gdxx*gdyy - gdxy**2)
guxy = idet*(gdxz*gdyz - gdzz*gdxy)
guxz = idet*(gdxy*gdyz - gdyy*gdxz)
guyz = idet*(gdxy*gdxz - gdxx*gdyz)
! Find first spatial derivatives of f.
dfx = idx*(ahfgrid(i+1,j,k) - ahfgrid(i-1,j,k))
dfy = idy*(ahfgrid(i,j+1,k) - ahfgrid(i,j-1,k))
dfz = idz*(ahfgrid(i,j,k+1) - ahfgrid(i,j,k-1))
! Raise indices in first derivatives.
dfux = guxx*dfx + guxy*dfy + guxz*dfz
dfuy = guxy*dfx + guyy*dfy + guyz*dfz
dfuz = guxz*dfx + guyz*dfy + guzz*dfz
! Find derivatives of metric using finite differences.
ddxxx = idx*(gxx(i+1,j,k) - gxx(i-1,j,k))
ddxyy = idx*(gyy(i+1,j,k) - gyy(i-1,j,k))
ddxzz = idx*(gzz(i+1,j,k) - gzz(i-1,j,k))
ddxxy = idx*(gxy(i+1,j,k) - gxy(i-1,j,k))
ddxxz = idx*(gxz(i+1,j,k) - gxz(i-1,j,k))
ddxyz = idx*(gyz(i+1,j,k) - gyz(i-1,j,k))
ddyxx = idy*(gxx(i,j+1,k) - gxx(i,j-1,k))
ddyyy = idy*(gyy(i,j+1,k) - gyy(i,j-1,k))
ddyzz = idy*(gzz(i,j+1,k) - gzz(i,j-1,k))
ddyxy = idy*(gxy(i,j+1,k) - gxy(i,j-1,k))
ddyxz = idy*(gxz(i,j+1,k) - gxz(i,j-1,k))
ddyyz = idy*(gyz(i,j+1,k) - gyz(i,j-1,k))
ddzxx = idz*(gxx(i,j,k+1) - gxx(i,j,k-1))
ddzyy = idz*(gyy(i,j,k+1) - gyy(i,j,k-1))
ddzzz = idz*(gzz(i,j,k+1) - gzz(i,j,k-1))
ddzxy = idz*(gxy(i,j,k+1) - gxy(i,j,k-1))
ddzxz = idz*(gxz(i,j,k+1) - gxz(i,j,k-1))
ddzyz = idz*(gyz(i,j,k+1) - gyz(i,j,k-1))
! Find second spatial derivatives of f.
d2fxx = (ahfgrid(i+1,j,k) - two*ahfgrid(i,j,k)
. + ahfgrid(i-1,j,k))*idx2
d2fyy = (ahfgrid(i,j+1,k) - two*ahfgrid(i,j,k)
. + ahfgrid(i,j-1,k))*idy2
d2fzz = (ahfgrid(i,j,k+1) - two*ahfgrid(i,j,k)
. + ahfgrid(i,j,k-1))*idz2
d2fxy = (ahfgrid(i+1,j+1,k) + ahfgrid(i-1,j-1,k)
. - ahfgrid(i+1,j-1,k) - ahfgrid(i-1,j+1,k))*idxy
d2fxz = (ahfgrid(i+1,j,k+1) + ahfgrid(i-1,j,k-1)
. - ahfgrid(i+1,j,k-1) - ahfgrid(i-1,j,k+1))*idxz
d2fyz = (ahfgrid(i,j+1,k+1) + ahfgrid(i,j-1,k-1)
. - ahfgrid(i,j+1,k-1) - ahfgrid(i,j-1,k+1))*idyz
! Find second covariant derivatives of f.
c2fxx = d2fxx - half*(dfux*ddxxx
. + dfuy*(two*ddxxy - ddyxx)
. + dfuz*(two*ddxxz - ddzxx))
c2fyy = d2fyy - half*(dfuy*ddyyy
. + dfux*(two*ddyxy - ddxyy)
. + dfuz*(two*ddyyz - ddzyy))
c2fzz = d2fzz - half*(dfuz*ddzzz
. + dfux*(two*ddzxz - ddxzz)
. + dfuy*(two*ddzyz - ddyzz))
c2fxy = d2fxy - half*(dfux*ddyxx + dfuy*ddxyy
. + dfuz*(ddxyz + ddyxz - ddzxy))
c2fxz = d2fxz - half*(dfux*ddzxx + dfuz*ddxzz
. + dfuy*(ddxyz + ddzxy - ddyxz))
c2fyz = d2fyz - half*(dfuy*ddzyy + dfuz*ddyzz
. + dfux*(ddyxz + ddzxy - ddxyz))
! 2 / m \ - 1
! Find: T0 = 1 / u = | d f d f |
! \ m /
T0 = dfx*dfux + dfy*dfuy + dfz*dfuz
if (T0.gt.zero) then
T0 = one/T0
else
betax(i,j,k) = zero
betay(i,j,k) = zero
betaz(i,j,k) = zero
goto 10
end if
! __2
! Find: T1 = \/ f
T1 = guxx*c2fxx + guyy*c2fyy + guzz*c2fzz
. + two*(guxy*c2fxy + guxz*c2fxz + guyz*c2fyz)
! __ __ a b 2
! Find: T2 = \/ \/ f d f d f / u
! a b
T2 = c2fxx*dfux**2 + c2fyy*dfuy**2 + c2fzz*dfuz**2
. + two*(c2fxy*dfux*dfuy + c2fxz*dfux*dfuz
. + c2fyz*dfuy*dfuz)
T2 = T2*T0
! Find: T3 = trK
T3 = guxx*kdxx + guyy*kdyy + guzz*kdzz
. + two*(guxy*kdxy + guxz*kdxz + guyz*kdyz)
! a b 2
! Find: T4 = K d f d f / u
! ab
T4 = kdxx*dfux**2 + kdyy*dfuy**2 + kdzz*dfuz**2
. + two*(kdxy*dfux*dfuy + kdxz*dfux*dfuz
. + kdyz*dfuy*dfuz)
T4 = T4*T0
! Find shift.
aux = alp(i,j,k)*(T3 - T4)/(T1 - T2)
betax(i,j,k) = aux*dfux
betay(i,j,k) = aux*dfuy
betaz(i,j,k) = aux*dfuz
10 continue
end do
end do
end do
! Boundaries.
if (CCTK_EQUALS(ahf_shiftbound,'extrap')) then
! X direction.
betax(1,:,:) = two*betax(2,:,:) - betax(3,:,:)
betay(1,:,:) = two*betay(2,:,:) - betay(3,:,:)
betaz(1,:,:) = two*betaz(2,:,:) - betaz(3,:,:)
betax(nx,:,:) = two*betax(nx-1,:,:) - betax(nx-2,:,:)
betay(nx,:,:) = two*betay(nx-1,:,:) - betay(nx-2,:,:)
betaz(nx,:,:) = two*betaz(nx-1,:,:) - betaz(nx-2,:,:)
! Y direction.
betax(:,1,:) = two*betax(:,2,:) - betax(:,3,:)
betay(:,1,:) = two*betay(:,2,:) - betay(:,3,:)
betaz(:,1,:) = two*betaz(:,2,:) - betaz(:,3,:)
betax(:,ny,:) = two*betax(:,ny-1,:) - betax(:,ny-2,:)
betay(:,ny,:) = two*betay(:,ny-1,:) - betay(:,ny-2,:)
betaz(:,ny,:) = two*betaz(:,ny-1,:) - betaz(:,ny-2,:)
! Z direction.
betax(:,:,1) = two*betax(:,:,2) - betax(:,:,3)
betay(:,:,1) = two*betay(:,:,2) - betay(:,:,3)
betaz(:,:,1) = two*betaz(:,:,2) - betaz(:,:,3)
betax(:,:,nz) = two*betax(:,:,nz-1) - betax(:,:,nz-2)
betay(:,:,nz) = two*betay(:,:,nz-1) - betay(:,:,nz-2)
betaz(:,:,nz) = two*betaz(:,:,nz-1) - betaz(:,:,nz-2)
end if
! ********************
! *** EXCISION ***
! ********************
if (ahf_shiftexcise.eq.1) then
#ifdef EXCISION_SIMPLEEXCISION
order_save = order
order = 1
call SimpleExcision(_CCTK_ARGUMENTS,x,y,z,r,mask,
. betax,betax_old,zero,one,'extrapolation')
call SimpleExcision(_CCTK_ARGUMENTS,x,y,z,r,mask,
. betay,betay_old,zero,one,'extrapolation')
call SimpleExcision(_CCTK_ARGUMENTS,x,y,z,r,mask,
. betaz,betaz_old,zero,one,'extrapolation')
order = order_save
#else
call CCTK_WARN(0,"You have not compiled with SimpleExcision")
#endif
end if
! ********************************************
! *** APPLY SYMMETRIES AND SYNCHRONIZE ***
! ********************************************
call CartSymGN(ierr,cctkGH,"admbase::shift")
call CCTK_SyncGroup(ierr,cctkGH,"admbase::shift")
! ***************
! *** END ***
! ***************
end subroutine AHFinder_shift
|
