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c /*@@
c @file DistortedBHIVP.F
c @date
c @author
c @desc
c
c @enddesc
c@@ */
#include "cctk.h"
#include "cctk_Parameters.h"
#include "cctk_Arguments.h"
c /*@@
c @routine DistortedBHIVP
c @date
c @author
c @desc
c
c @enddesc
c @calls
c @calledby
c @history
c
c @endhistory
c@@ */
c Need include file from Einstein
#include "CactusEinstein/Einstein/src/Einstein.h"
subroutine DistortedBHIVP(CCTK_FARGUMENTS)
implicit none
DECLARE_CCTK_FARGUMENTS
DECLARE_CCTK_PARAMETERS
c Perhaps this and others should go into cctk.h
integer CCTK_Equals
real*8 :: deta,dq,dphi
real*8, allocatable :: ac(:,:,:),ae(:,:,:),aw(:,:,:),an(:,:,:),
$ as(:,:,:),aq(:,:,:),ab(:,:,:),rhs(:,:,:),
$ qf(:,:,:),qfetaeta(:,:,:),qfqq(:,:,:),qfphi(:,:,:),
$ qfphiphi(:,:,:),
$ psisph(:,:,:),psiprim(:,:,:),detapsisph(:,:,:),
$ dqpsisph(:,:,:),dphipsisph(:,:,:),detaetapsisph(:,:,:),
$ detaqpsisph(:,:,:),detaphipsisph(:,:,:),dqqpsisph(:,:,:),
$ dqphipsisph(:,:,:),dphiphipsisph(:,:,:)
real*8, allocatable :: etagrd(:),qgrd(:),phigrd(:)
real*8 o1,o2,o3,o4,o5,o6,o7,o8,o9,o10,o11,o12,o13,o14,o15,o16,o17,
$ o18,o19,o20,o21,o22,o23,o24,o25,o26,o27,o28,o29,o30,o31,o32,
$ o33,o34,o35,o36,o37,o38,o39,o40,o41,o42,o43,o44,o45,o46,o47,
$ o48,o49,o50,o51,o52,o53,o54,o55,o56,o57,o58,o59,o60,o61,o62,
$ o63,o64,o65,o66,o67
real*8 rmax,adm
real*8,parameter :: dbh_eps = 1.0d-9
real*8 pi
integer :: ne, nq, np
integer :: nx,ny,nz
integer i,j,k,ier,nquads,nocts,order
integer npoints,handle,ierror
conformal_state = CONFORMAL_METRIC
pi = 4.0d0*atan(1.0d0)
c DONT use integer*4
ne = neta
nq = ntheta
np = nphi
c Set up the grid spacings
nx = cctk_lsh(1)
ny = cctk_lsh(2)
nz = cctk_lsh(3)
c Distorted Schwarzchild BH parameters
print *,"Brill wave + Distorted BH solve"
write(*,123)amp,eta0,c,sigma,n
print*,'etamax=',etamax
123 format(1x, 'Pars: amp',f8.5,' eta0',f8.5,' c',f8.5,' sigma',f8.5,' n',i3)
c Sovle on this sized cartesian grid
c 3D grid size NE x NT x NP
c Add 2 zones for eta coordinate and 4 for theta
c and phi coordenate.
ne = ne + 2
nq = nq + 2
np = np + 2
c
allocate(ac(ne,nq,np),ae(ne,nq,np),aw(ne,nq,np),an(ne,nq,np),
$ as(ne,nq,np),aq(ne,nq,np),ab(ne,nq,np),rhs(ne,nq,np),
$ qf(ne,nq,np),qfetaeta(ne,nq,np),qfqq(ne,nq,np),
$ qfphi(ne,nq,np),qfphiphi(ne,nq,np),
$ psisph(ne,nq,np),psiprim(ne,nq,np),detapsisph(ne,nq,np),
$ dqpsisph(ne,nq,np),dphipsisph(ne,nq,np),
$ detaetapsisph(ne,nq,np),detaqpsisph(ne,nq,np),
$ detaphipsisph(ne,nq,np),dqqpsisph(ne,nq,np),
$ dqphipsisph(ne,nq,np),dphiphipsisph(ne,nq,np))
allocate(etagrd(ne),qgrd(nq),phigrd(np))
c
c Initialize some array
c
nocts = 4
nquads = 2
dphi = nocts*0.5*pi/(np-2)
dq = nquads*0.5*pi/(nq-2)
deta = etamax/(ne-3)
do k = 1,np
phigrd(k) = (k-1.5)*dphi
enddo
do j = 1,nq
qgrd(j) = (j-1.5)*dq
enddo
do i=1,ne
etagrd(i) = (i-2)*deta
enddo
c
c Initialize q-function and its derivatives: should be generalized
c
do k = 1,np
do j = 1,nq
do i = 1,ne
#include "qfunc.x"
enddo
enddo
enddo
c
c Initialize psi to the Schwarzschild solution:
c
psiprim = 0.
do k = 1,np
do j = 1,nq
do i = 1,ne
psisph(i,j,k) = 2.*cosh(0.5*etagrd(i))
enddo
enddo
enddo
c
c Initialize stencil coefficients:
c
ac = 0.
ae = 0.
aw = 0.
an = 0.
as = 0.
aq = 0.
ab = 0.
rhs = 0.
c
do k = 2,np-1
do j = 2,nq-1
do i = 2,ne-1
ac(i,j,k) = -2./deta**2-2./dq**2-2.*exp(2.*
& qf(i,j,k))/(dphi**2*
& sin(qgrd(j))**2)+0.25*
& (qfetaeta(i,j,k)+qfqq(i,j,k)+2.*
& exp(2.*qf(i,j,k))*qfphiphi(i,j,k)/
& sin(qgrd(j))**2+3.*exp(2.*
& qf(i,j,k))*qfphi(i,j,k)**2/
& sin(qgrd(j))**2-1.)
ae(i,j,k) = 1./deta**2
aw(i,j,k) = 1./deta**2
an(i,j,k) = 1./dq**2+0.5/(dq*tan(qgrd(j)))
as(i,j,k) = 1./dq**2-0.5/(dq*tan(qgrd(j)))
aq(i,j,k) = exp(2.*qf(i,j,k))/(dphi**2*
& sin(qgrd(j))**2)+exp(2.*
& qf(i,j,k))*qfphi(i,j,k)/(dphi*
& sin(qgrd(j))**2)
ab(i,j,k) = exp(2.*qf(i,j,k))/(dphi**2*
& sin(qgrd(j))**2)-exp(2.*
& qf(i,j,k))*qfphi(i,j,k)/(dphi*
& sin(qgrd(j))**2)
rhs(i,j,k) = -0.25*(qfetaeta(i,j,k)+
& qfqq(i,j,k)+2.*exp(2.*qf(i,j,k))*
& qfphiphi(i,j,k)/sin(qgrd(j))**2+3.*
& exp(2.*qf(i,j,k))*qfphi(i,j,k)**2/
& sin(qgrd(j))**2)*psisph(i,j,k)
enddo
enddo
enddo
c
c Apply boundary conditions to the faces of the cube:
c
c i=2:
do k = 3,np-2
do j = 3,nq-2
ae(2,j,k) = ae(2,j,k) + aw(2,j,k)
aw(2,j,k) = 0.
c
c i=ne-1:
ac(ne-1,j,k) = ac(ne-1,j,k)+4.*ae(ne-1,j,k)/(3.+deta)
aw(ne-1,j,k) = aw(ne-1,j,k) - ae(ne-1,j,k)/(3.+deta)
ae(ne-1,j,k) = 0.
enddo
enddo
c
c j=2:
do k = 3,np-2
do i = 3,ne-2
ac(i,2,k) = ac(i,2,k) + as(i,2,k)
as(i,2,k) = 0.
c
c j=nq-1:
ac(i,nq-1,k) = ac(i,nq-1,k) + an(i,nq-1,k)
an(i,nq-1,k) = 0.
enddo
enddo
c
c k=2:
do j = 3,nq-2
do i = 3,ne-2
ac(i,j,2) = ac(i,j,2) + ab(i,j,2)
ab(i,j,2) = 0.
c
c k=np-1:
ac(i,j,np-1) = ac(i,j,np-1) + aq(i,j,np-1)
aq(i,j,np-1) = 0.
enddo
enddo
c
c Apply boundary conditions to the edges of the cube:
c
c i=2, j=2:
do k = 3,np-2
ae(2,2,k) = ae(2,2,k) + aw(2,2,k)
ac(2,2,k) = ac(2,2,k) + as(2,2,k)
aw(2,2,k) = 0.
as(2,2,k) = 0.
c
c i=ne-1, j=2:
aw(ne-1,2,k) = aw(ne-1,2,k) - ae(ne-1,2,k)/(3.+deta)
ac(ne-1,2,k) = ac(ne-1,2,k) + as(ne-1,2,k) +
& 4.*ae(ne-1,2,k)/(3.+deta)
ae(ne-1,2,k) = 0.
as(ne-1,2,k) = 0.
c
c i=2, j=nq-1:
ae(2,nq-1,k) = ae(2,nq-1,k) + aw(2,nq-1,k)
ac(2,nq-1,k) = ac(2,nq-1,k) + an(2,nq-1,k)
aw(2,nq-1,k) = 0.
an(2,nq-1,k) = 0.
c
c i=ne-1, j=nq-1:
aw(ne-1,nq-1,k) = aw(ne-1,nq-1,k) - ae(ne-1,nq-1,k)/
& (3.+deta)
ac(ne-1,nq-1,k) = ac(ne-1,nq-1,k) + an(ne-1,nq-1,k) +
& 4.*ae(ne-1,nq-1,k)/(3.+deta)
ae(ne-1,nq-1,k) = 0.
an(ne-1,nq-1,k) = 0.
enddo
c
c i=2, k=2:
do j = 3,nq-2
ae(2,j,2) = ae(2,j,2) + aw(2,j,2)
ac(2,j,2) = ac(2,j,2) + ab(2,j,2)
aw(2,j,2) = 0.
ab(2,j,2) = 0.
c
c i=ne-1, k=2:
aw(ne-1,j,2) = aw(ne-1,j,2) - ae(ne-1,j,2)/(3.+deta)
ac(ne-1,j,2) = ac(ne-1,j,2) + ab(ne-1,j,2) +
& 4.*ae(ne-1,j,2)/(3.+deta)
ae(ne-1,j,2) = 0.
ab(ne-1,j,2) = 0.
c
c i=2, k=np-1:
ae(2,j,np-1) = ae(2,j,np-1) + aw(2,j,np-1)
ac(2,j,np-1) = ac(2,j,np-1) + aq(2,j,np-1)
aw(2,j,np-1) = 0.
aq(2,j,np-1) = 0.
c
c i=ne-1, k=np-1:
aw(ne-1,j,np-1) = aw(ne-1,j,np-1) - ae(ne-1,j,np-1)/
& (3.+deta)
ac(ne-1,j,np-1) = ac(ne-1,j,np-1) + aq(ne-1,j,np-1) +
& 4.*ae(ne-1,j,np-1)/(3.+deta)
ae(ne-1,j,np-1) = 0.
aq(ne-1,j,np-1) = 0.
enddo
c
c j=2, k=2:
do i = 3,ne-2
ac(i,2,2) = ac(i,2,2) + as(i,2,2) + ab(i,2,2)
as(i,2,2) = 0.
ab(i,2,2) = 0.
c
c j=nq-1, k=2:
ac(i,nq-1,2) = ac(i,nq-1,2) + an(i,nq-1,2) +
& ab(i,nq-1,2)
an(i,nq-1,2) = 0.
ab(i,nq-1,2) = 0.
c
c j=2, k=np-1:
ac(i,2,np-1) = ac(i,2,np-1) + as(i,2,np-1) +
& aq(i,2,np-1)
as(i,2,np-1) = 0.
aq(i,2,np-1) = 0.
c
c j=nq-1, k=np-1:
ac(i,nq-1,np-1) = ac(i,nq-1,np-1) + an(i,nq-1,np-1) +
& aq(i,nq-1,np-1)
an(i,nq-1,np-1) = 0.
aq(i,nq-1,np-1) = 0.
enddo
c
c Apply boundary conditions to the corners of the cube:
c
c i=2, j=2, k=2:
ae(2,2,2) = ae(2,2,2) + aw(2,2,2)
ac(2,2,2) = ac(2,2,2) + as(2,2,2) + ab(2,2,2)
aw(2,2,2) = 0.
as(2,2,2) = 0.
ab(2,2,2) = 0.
c
c i=ne-1, j=2, k=2:
aw(ne-1,2,2) = aw(ne-1,2,2) - ae(ne-1,2,2)/(3.+deta)
ac(ne-1,2,2) = ac(ne-1,2,2) + as(ne-1,2,2) + ab(ne-1,2,2) +
& 4.*ae(ne-1,2,2)/(3.+deta)
ae(ne-1,2,2) = 0.
as(ne-1,2,2) = 0.
ab(ne-1,2,2) = 0.
c
c i=2, j=nq-1, k=2:
ae(2,nq-1,2) = ae(2,nq-1,2) + aw(2,nq-1,2)
ac(2,nq-1,2) = ac(2,nq-1,2) + an(2,nq-1,2) + ab(2,nq-1,2)
aw(2,nq-1,2) = 0.
an(2,nq-1,2) = 0.
ab(2,nq-1,2) = 0.
c
c i=2, j=2, k=np-1:
ae(2,2,np-1) = ae(2,2,np-1) + aw(2,2,np-1)
ac(2,2,np-1) = ac(2,2,np-1) + as(2,2,np-1) + aq(2,2,np-1)
aw(2,2,np-1) = 0.
as(2,2,np-1) = 0.
aq(2,2,np-1) = 0.
c
c i=ne-1, j=nq-1, k=2:
aw(ne-1,nq-1,2) = aw(ne-1,nq-1,2) - ae(ne-1,nq-1,2)/(3.+deta)
ac(ne-1,nq-1,2) = ac(ne-1,nq-1,2) + an(ne-1,nq-1,2) + ab(ne-1,nq-1,2) +
& 4.*ae(ne-1,nq-1,2)/(3.+deta)
ae(ne-1,nq-1,2) = 0.
an(ne-1,nq-1,2) = 0.
ab(ne-1,nq-1,2) = 0.
c
c i=ne-1, j=2, k=np-1:
aw(ne-1,2,np-1) = aw(ne-1,2,np-1) - ae(ne-1,2,np-1)/(3.+deta)
ac(ne-1,2,np-1) = ac(ne-1,2,np-1) + as(ne-1,2,np-1) + aq(ne-1,2,np-1) +
& 4.*ae(ne-1,2,np-1)/(3.+deta)
ae(ne-1,2,np-1) = 0.
as(ne-1,2,np-1) = 0.
aq(ne-1,2,np-1) = 0.
c
c i=2, j=nq-1, k=np-1:
ae(2,nq-1,np-1) = ae(2,nq-1,np-1) + aw(2,nq-1,np-1)
ac(2,nq-1,np-1) = ac(2,nq-1,np-1) + an(2,nq-1,np-1) + aq(2,nq-1,np-1)
aw(2,nq-1,np-1) = 0.
an(2,nq-1,np-1) = 0.
aq(2,nq-1,np-1) = 0.
c
c i=ne-1, j=nq-1, k=np-1:
aw(ne-1,nq-1,np-1) = aw(ne-1,nq-1,np-1) - ae(ne-1,nq-1,np-1)/(3.+deta)
ac(ne-1,nq-1,np-1) = ac(ne-1,nq-1,np-1) + an(ne-1,nq-1,np-1) +
& aq(ne-1,nq-1,np-1) + 4.*ae(ne-1,nq-1,np-1)/(3.+deta)
ae(ne-1,nq-1,np-1) = 0.
an(ne-1,nq-1,np-1) = 0.
aq(ne-1,nq-1,np-1) = 0.
c
c Solve for psi:
c
call bicgst3d(ac,ae,aw,an,as,aq,ab,psiprim,rhs,dbh_eps,rmax,ier,
$ ne,nq,np)
c
if (rmax.gt.dbh_eps) then
write(*,*) '***WARNING: bicgst3d did not converge.'
endif
if (ier.eq.-1) then
write(*,*) '***WARNING: ier=-1'
endif
print *,'psiprim = ',maxval(psiprim),' ',minval(psiprim)
c
c Now, apply boundary conditions to psiprim:
c
do k = 1,np
do j = 1,nq
psiprim(1,j,k) = psiprim(3,j,k)
psiprim(ne,j,k) = (4.*psiprim(ne-1,j,k)-psiprim(ne-2,j,k))/
$ (3.+deta)
enddo
enddo
do k = 1,np
do i = 1,ne
psiprim(i,1,k) = psiprim(i,2,k)
psiprim(i,nq,k) = psiprim(i,nq-1,k)
enddo
enddo
do j = 1,nq
do i = 1,ne
psiprim(i,j,1) = psiprim(i,j,2)
psiprim(i,j,np) = psiprim(i,j,np-1)
enddo
enddo
c
c Here, compute the derivatives of the spherical conformal factor
c
c goto 110
do k = 1,np
do j = 1,nq
do i = 2,ne-1
detapsisph(i,j,k)=0.5*(psiprim(i+1,j,k)-psiprim(i-1,j,k))
$ /deta + sinh(0.5*etagrd(i))
enddo
detapsisph(1,j,k) = -detapsisph(3,j,k)
enddo
enddo
c
do k = 1,np
do j = 2,nq-1
do i = 1,ne
dqpsisph(i,j,k)=0.5*(psiprim(i,j+1,k)-psiprim(i,j-1,k))/
$ dq
enddo
enddo
do i = 1,ne
dqpsisph(i,1,k) = -dqpsisph(i,2,k)
dqpsisph(i,nq,k) = -dqpsisph(i,nq-1,k)
enddo
enddo
c
do k = 2,np-1
do j = 1,nq
do i = 1,ne
dphipsisph(i,j,k)=0.5*(psiprim(i,j,k+1)-psiprim(i,j,k-1))
$ /dphi
enddo
enddo
enddo
do j = 1,nq
do i = 1,ne
dphipsisph(i,j,1) = -dphipsisph(i,j,2)
dphipsisph(i,j,np) = -dphipsisph(i,j,np-1)
enddo
enddo
c
do k = 1,np
do j = 1,nq
do i = 2,ne-1
detaetapsisph(i,j,k)=(psiprim(i+1,j,k)-2.*psiprim(i,j,k)+
& psiprim(i-1,j,k))/deta**2+sqrt(0.25)*
& cosh(0.5*etagrd(i))
enddo
detaetapsisph(1,j,k) = detaetapsisph(3,j,k)
enddo
enddo
c
do k = 1,np
do j = 2,nq-1
do i = 1,ne
detaqpsisph(i,j,k)=0.5*(detapsisph(i,j+1,k)-detapsisph(i,
$ j-1,k))/dq
enddo
enddo
do i = 1,ne
detaqpsisph(i,1,k) = -detaqpsisph(i,2,k)
detaqpsisph(i,nq,k) = -detaqpsisph(i,nq-1,k)
enddo
enddo
c
do k = 2,np-1
do j = 1,nq
do i = 1,ne
detaphipsisph(i,j,k)=0.5*(detapsisph(i,j,k+1)-detapsisph(
$ i,j,k-1))/dphi
enddo
enddo
enddo
do j = 1,nq
do i = 1,ne
detaphipsisph(i,j,1) = -detaphipsisph(i,j,2)
detaphipsisph(i,j,np) = -detaphipsisph(i,j,np-1)
enddo
enddo
c
do k = 1,np
do j = 2,nq-1
do i = 1,ne
dqqpsisph(i,j,k)=0.5*(dqpsisph(i,j+1,k)-dqpsisph(i,j-1,k))/
$ dq
enddo
enddo
do i = 1,ne
dqqpsisph(i,1,k) = dqqpsisph(i,2,k)
dqqpsisph(i,nq,k) = dqqpsisph(i,nq-1,k)
enddo
enddo
c
do k = 2,np-1
do j = 1,nq
do i = 1,ne
dqphipsisph(i,j,k)=0.5*(dqpsisph(i,j,k+1)-dqpsisph(i,j,k-1))/
$ dphi
enddo
enddo
enddo
do j = 1,nq
do i = 1,ne
dqphipsisph(i,j,1) = -dqphipsisph(i,j,2)
dqphipsisph(i,j,np) = -dqphipsisph(i,j,np-1)
enddo
enddo
c
do k = 2,np-1
do j = 1,nq
do i = 1,ne
dphiphipsisph(i,j,k)=0.5*(dphipsisph(i,j,k+1)-
$ dphipsisph(i,j,k-1))/dphi
enddo
enddo
enddo
do j = 1,nq
do i = 1,ne
dphiphipsisph(i,j,1) = dphiphipsisph(i,j,2)
dphiphipsisph(i,j,np) = dphiphipsisph(i,j,np-1)
enddo
enddo
c
do k = 1,np
do j = 1,nq
psisph(:,j,k)=psiprim(:,j,k)+2.0*cosh(0.5*etagrd)
enddo
enddo
c
c Now compute on the Cartesian coordinate.
c
c Compute eta,q,phi at the each points of cartesian grid
eta = 0.5d0*dlog(x**2+y**2+z**2)
abseta = abs(eta)
q = datan2(sqrt(x**2+y**2),z)
phi = datan2(y,x)
do k = 1,nz
do j = 1,ny
do i = 1,nx
if(eta(i,j,k) .lt. 0)then
sign_eta(i,j,k) = -1.0d0
else
sign_eta(i,j,k) = 1.0d0
endif
enddo
enddo
enddo
call CCTK_InterpHandle (handle, "simple_local")
npoints = nx*ny*nz
call CCTK_Interp (ierror,cctkGH,handle,npoints,3,10,10,
$ ne,nq,np,abseta,q,phi,
$ CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,
$ etagrd(1),qgrd(1),phigrd(1)-pi,deta,dq,dphi,
$ psisph,detapsisph,dqpsisph,dphipsisph,detaetapsisph,
$ detaqpsisph,detaphipsisph,dqqpsisph,dqphipsisph,dphiphipsisph,
$ CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,
$ CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,
$ CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,
$ CCTK_VARIABLE_REAL,
$ psi3d,detapsi3d,dqpsi3d,dphipsi3d,detaetapsi3d,detaqpsi3d,
$ detaphipsi3d,dqqpsi3d,dqphipsi3d,dphiphipsi3d,
$ CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,
$ CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,
$ CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,CCTK_VARIABLE_REAL,
$ CCTK_VARIABLE_REAL)
psi = psi3d*exp(-0.5*eta)
detapsi3d = sign_eta*detapsi3d
detaqpsi3d = sign_eta*detaqpsi3d
detaphipsi3d = sign_eta*detaphipsi3d
do k=1,nz
do j=1,ny
do i=1,nx
c psix = \partial psi / \partial x / psi
#include "psi_1st_deriv.x"
c psixx = \partial^2\psi / \partial x^2 / psi
#include "psi_2nd_deriv.x"
enddo
enddo
enddo
c Conformal metric
c gxx = ...
c Derivatives of the metric
c dxgxx = 1/2 \partial gxx / \partial x
c
do k=1,nz
do j=1,ny
do i=1,nx
#include "gij.x"
enddo
enddo
enddo
c Courvature
kxx = 0.0d0
kxy = 0.0d0
kxz = 0.0d0
kyy = 0.0d0
kyz = 0.0d0
kzz = 0.0d0
110 continue
c Set ADM mass
i = ne-15
adm = 0.0
do k=2,np-1
do j=2,nq-1
adm=adm+(psisph(i,j,k)-(psisph(i+1,j,k)-psisph(i-1,j,k))/
$ deta)*exp(0.5*etagrd(i))
enddo
enddo
adm=adm/(nq-2)/(np-2)
print *,'ADM mass: ',adm
if (CCTK_Equals(initial_lapse,"schwarz")==1) then
write (*,*)"Initial with schwarzschild-like lapse"
write (*,*)"using alp = (2.*r - adm)/(2.*r+adm)."
alp = (2.*r - adm)/(2.*r+adm)
endif
deallocate(ac,ae,aw,an,as,aq,ab,rhs,
$ qf,qfetaeta,qfqq,qfphi,qfphiphi,
$ psisph,psiprim,detapsisph,dqpsisph,dphipsisph,detaetapsisph,
$ detaqpsisph,detaphipsisph,dqqpsisph,dqphipsisph,
$ dphiphipsisph)
deallocate(etagrd,qgrd,phigrd)
return
end
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