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AMSS-NCKU/AMSS_NCKU_source/getnp4old.f90
CGH0S7 f2fc9af70e asc26 amss-ncku initialized
2026-01-13 15:01:15 +08:00

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#include "macrodef.fh"
!-----------------------------------------------------------------------------
!
! compute rhw Newman-Penrose Weyl scalar Psi4
! for BSSN dynamical variables
!
!-----------------------------------------------------------------------------
subroutine getnp4old(ex, X, Y, Z, chi, trK, &
dxx, gxy, gxz, dyy, gyz, dzz, &
Axx, Axy, Axz, Ayy, Ayz, Azz, &
Gmx, Gmy, Gmz, &
Lap, Sfx, Sfy, Sfz,Rpsi4,Ipsi4, symmetry)
implicit none
!~~~~~~> Input parameters:
integer,intent(in ):: ex(1:3),symmetry
real*8, intent(in ):: X(1:ex(1)),Y(1:ex(2)),Z(1:ex(3))
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: chi,trK
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: dxx,gxy,gxz,dyy,gyz,dzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Axx,Axy,Axz,Ayy,Ayz,Azz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Gmx,Gmy,Gmz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Lap,Sfx,Sfy,Sfz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: Rpsi4,Ipsi4
!~~~~~~> Other variables:
real*8, dimension(ex(1),ex(2),ex(3)) :: gupxx,gupxy,gupxz
real*8, dimension(ex(1),ex(2),ex(3)) :: gupyy,gupyz,gupzz
real*8, dimension(ex(1),ex(2),ex(3)) :: ep4phi,alpn1
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gyy,gzz
real*8, dimension(ex(1),ex(2),ex(3)) :: phi,phix,phiy,phiz
real*8, dimension(ex(1),ex(2),ex(3)) :: phixx,phixy,phixz,phiyy,phiyz,phizz
real*8, dimension(ex(1),ex(2),ex(3)) :: tRxyxy, tRxyxz, tRxyyz, tRxzxz, tRxzyz, tRyzyz
real*8, dimension(ex(1),ex(2),ex(3)) :: Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz
real*8, dimension(ex(1),ex(2),ex(3)) :: vx,vy,vz,ux,uy,uz,wx,wy,wz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamxxx, Gamxxy, Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamxyy, Gamxyz, Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamyxx, Gamyxy, Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamyyy, Gamyyz, Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamzxx, Gamzxy, Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamzyy, Gamzyz, Gamzzz
real*8, dimension(ex(1),ex(2),ex(3)) :: tRxx,tRxy,tRxz,tRyy,tRyz,tRzz
real*8, dimension(ex(1),ex(2),ex(3)) :: Rxx, Rxy, Rxz, Ryy, Ryz, Rzz
real*8, dimension(ex(1),ex(2),ex(3)) :: Kxx,Kxy,Kxz,Kyy,Kyz,Kzz
!D_i K_jk ---> DKijk
real*8, dimension(ex(1),ex(2),ex(3)) :: DKxxx,DKxxy,DKxxz,DKxyy,DKxyz,DKxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: DKyxx,DKyxy,DKyxz,DKyyy,DKyyz,DKyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: DKzxx,DKzxy,DKzxz,DKzyy,DKzyz,DKzzz
! Aij,k --> stored as Aijk
real*8, dimension(ex(1),ex(2),ex(3))::Axxx,Axxy,Axxz
real*8, dimension(ex(1),ex(2),ex(3))::Axyx,Axyy,Axyz
real*8, dimension(ex(1),ex(2),ex(3))::Axzx,Axzy,Axzz
real*8, dimension(ex(1),ex(2),ex(3))::Ayyx,Ayyy,Ayyz
real*8, dimension(ex(1),ex(2),ex(3))::Ayzx,Ayzy,Ayzz
real*8, dimension(ex(1),ex(2),ex(3))::Azzx,Azzy,Azzz
! trK,i
real*8, dimension(ex(1),ex(2),ex(3))::Kx,Ky,Kz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamxxx,ass_Gamxxy,ass_Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamxyy,ass_Gamxyz,ass_Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamyxx,ass_Gamyxy,ass_Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamyyy,ass_Gamyyz,ass_Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamzxx,ass_Gamzxy,ass_Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamzyy,ass_Gamzyz,ass_Gamzzz
! first order partial derivative of metric
real*8, dimension(ex(1),ex(2),ex(3)) :: gxxx,gxxy,gxxz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxyx,gxyy,gxyz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxzx,gxzy,gxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gyyx,gyyy,gyyz
real*8, dimension(ex(1),ex(2),ex(3)) :: gyzx,gyzy,gyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gzzx,gzzy,gzzz
! second order partial derivative of metric
real*8, dimension(ex(1),ex(2),ex(3)) :: gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz
real*8, parameter :: F1o4=2.5d-1,ONE=1.d0,TWO=2.d0,FOUR=4.d0
real*8, parameter :: SYM = 1.D0, ANTI= - 1.D0
phi = -0.25d0*dlog(chi+ONE)
!~~~~~~
gxx = dxx + ONE
gyy = dyy + ONE
gzz = dzz + ONE
! invert tilted metric
gupzz = gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
gupxx = ( gyy * gzz - gyz * gyz ) / gupzz
gupxy = - ( gxy * gzz - gyz * gxz ) / gupzz
gupxz = ( gxy * gyz - gyy * gxz ) / gupzz
gupyy = ( gxx * gzz - gxz * gxz ) / gupzz
gupyz = - ( gxx * gyz - gxy * gxz ) / gupzz
gupzz = ( gxx * gyy - gxy * gxy ) / gupzz
alpn1 = Lap + ONE
ep4phi = dexp( FOUR * Phi )
!~~~~~~>
call d1metric(ex,X,Y,Z, &
dxx ,gxy ,gxz ,dyy ,gyz ,dzz , &
gxxx,gxyx,gxzx,gyyx,gyzx,gzzx, &
gxxy,gxyy,gxzy,gyyy,gyzy,gzzy, &
gxxz,gxyz,gxzz,gyyz,gyzz,gzzz, symmetry)
call d2metric(ex,X,Y,Z, &
dxx, gxy, gxz, dyy, gyz, dzz, &
gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz, &
gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz, &
gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz, &
gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz, &
gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz, &
gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz, symmetry)
call kind1_connection(ex, gxxx,gxyx,gxzx,gyyx,gyzx,gzzx, &
gxxy,gxyy,gxzy,gyyy,gyzy,gzzy, &
gxxz,gxyz,gxzz,gyyz,gyzz,gzzz, &
ass_Gamxxx, ass_Gamxxy, ass_Gamxxz,&
ass_Gamxyy, ass_Gamxyz, ass_Gamxzz,&
ass_Gamyxx, ass_Gamyxy, ass_Gamyxz,&
ass_Gamyyy, ass_Gamyyz, ass_Gamyzz,&
ass_Gamzxx, ass_Gamzxy, ass_Gamzxz,&
ass_Gamzyy, ass_Gamzyz, ass_Gamzzz)
call kind2_connection(ex, gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
ass_Gamxxx, ass_Gamxxy, ass_Gamxxz, &
ass_Gamxyy, ass_Gamxyz, ass_Gamxzz, &
ass_Gamyxx, ass_Gamyxy, ass_Gamyxz, &
ass_Gamyyy, ass_Gamyyz, ass_Gamyzz, &
ass_Gamzxx, ass_Gamzxy, ass_Gamzxz, &
ass_Gamzyy, ass_Gamzyz, ass_Gamzzz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz)
!~~~~~~> derivs of conformal factor
call fderivs(ex,phi,phix,phiy,phiz,X,Y,Z,SYM,SYM,SYM,Symmetry,0)
call fdderivs(ex,phi,phixx,phixy,phixz,phiyy,phiyz,phizz,X,Y,Z, &
SYM,SYM,SYM,symmetry,0)
call xcov_deriv(ex, phix, phiy, phiz, &
phixx, phixy, phixz, phiyy, phiyz, phizz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz)
!~~~~~~> get spatial Riemann curvature
call adm_riemann(ex, gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz, &
gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz, &
gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz, &
gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz, &
gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz, &
gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz, &
ass_Gamxxx,ass_Gamxxy,ass_Gamxxz, &
ass_Gamxyy,ass_Gamxyz,ass_Gamxzz, &
ass_Gamyxx,ass_Gamyxy,ass_Gamyxz, &
ass_Gamyyy,ass_Gamyyz,ass_Gamyzz, &
ass_Gamzxx,ass_Gamzxy,ass_Gamzxz, &
ass_Gamzyy,ass_Gamzyz,ass_Gamzzz, &
tRxyxy, tRxyxz, tRxyyz, tRxzxz, tRxzyz, tRyzyz)
call get_physical_riemann(ex, ep4phi, &
dxx , gxy , gxz , dyy , gyz , dzz , &
gupxx , gupxy , gupxz , gupyy , gupyz , gupzz , &
phix , phiy , phiz , &
phixx , phixy , phixz , phiyy , phiyz , phizz , &
tRxyxy, tRxyxz, tRxyyz, tRxzxz, tRxzyz, tRyzyz, &
Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz)
!~~~~~~> get spatial Ricci tensor
call adm_ricci(ex, gupxx, gupxy, gupxz, gupyy, gupyz, gupzz, &
tRxyxy,tRxyxz,tRxyyz,tRxzxz,tRxzyz,tRyzyz, &
tRxx, tRxy, tRxz, tRyy, tRyz, tRzz)
call get_physical_ricci(ex,dxx,gxy,gxz,dyy,gyz,dzz,phix,phiy,phiz, &
phixx,phixy,phixz,phiyy,phiyz,phizz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
tRxx, tRxy, tRxz, tRyy, tRyz, tRzz, &
Rxx, Rxy, Rxz, Ryy, Ryz, Rzz)
!~~~~~~> get the real spatial extrinsic curvature
call get_physical_k(ex, phi, trK, dxx, gxy, gxz, dyy, gyz, dzz, &
Axx, Axy, Axz, Ayy, Ayz, Azz, &
Kxx, Kxy, Kxz, Kyy, Kyz, Kzz)
!~~~~~~> derivs of trace of extrinsic curvature
call fderivs(ex,trK, Kx, Ky, Kz,X,Y,Z,SYM,SYM,SYM,Symmetry,0)
!~~~~~~> derivs of tilde extrinsic curvature
call fderivs(ex,Axx,Axxx,Axxy,Axxz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0)
call fderivs(ex,Axy,Axyx,Axyy,Axyz,X,Y,Z,ANTI,ANTI,SYM ,Symmetry,0)
call fderivs(ex,Axz,Axzx,Axzy,Axzz,X,Y,Z,ANTI,SYM ,ANTI,Symmetry,0)
call fderivs(ex,Ayy,Ayyx,Ayyy,Ayyz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0)
call fderivs(ex,Ayz,Ayzx,Ayzy,Ayzz,X,Y,Z,SYM ,ANTI,ANTI,Symmetry,0)
call fderivs(ex,Azz,Azzx,Azzy,Azzz,X,Y,Z,SYM ,SYM ,SYM ,Symmetry,0)
!~~~~~~> derivs of extrinsic curvature, Kij
call get_diff_physical_k(ex, phi, trK, Kx, Ky, Kz, phix, phiy, phiz, &
dxx, gxy, gxz, dyy, gyz, dzz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
Axx, Axy, Axz, Ayy, Ayz, Azz, &
Axxx, Axxy, Axxz, Axyx, Axyy, Axyz, &
Axzx, Axzy, Axzz, Ayyx, Ayyy, Ayyz, &
Ayzx, Ayzy, Ayzz, Azzx, Azzy, Azzz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz, &
Kxx, Kxy, Kxz, Kyy, Kyz, Kzz, &
DKxxx, DKxxy, DKxxz, DKxyy, DKxyz, DKxzz, &
DKyxx, DKyxy, DKyxz, DKyyy, DKyyz, DKyzz, &
DKzxx, DKzxy, DKzxz, DKzyy, DKzyz, DKzzz)
!~~~~~~> get the Gram-Schmidt orthonormalize triad coordinate
#if (tetradtype == 0)
call get_triad0(ex, X, Y, Z, ep4phi, gxx, gxy, gxz, gyy, gyz, gzz, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
#elif (tetradtype == 1)
call get_triad1(ex, X, Y, Z, ep4phi, gxx, gxy, gxz, gyy, gyz, gzz, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
#elif (tetradtype == 2)
call get_triad2(ex, X, Y, Z, ep4phi, gxx, gxy, gxz, gyy, gyz, gzz, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
#endif
!~~~~~~> compute the Newnamm-Penrose psi4 which split real and image part
ep4phi = ONE / ep4phi
call bssn_compute_psi4(ex,ep4phi, alpn1, Sfx, Sfy, Sfz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
vx,vy,vz,ux,uy,uz,wx,wy,wz, &
trK,Kxx,Kxy,Kxz,Kyy,Kyz,Kzz, &
Rxyxy,Rxyxz,Rxyyz,Rxzxz,Rxzyz,Ryzyz, &
Rxx, Rxy, Rxz, Ryy, Ryz, Rzz, &
DKxxx,DKxxy,DKxxz,DKxyy,DKxyz,DKxzz, &
DKyxx,DKyxy,DKyxz,DKyyy,DKyyz,DKyzz, &
DKzxx,DKzxy,DKzxz,DKzyy,DKzyz,DKzzz, Rpsi4, Ipsi4)
return
end subroutine getnp4old
!-----------------------------------------------------------------------------------
! for shell
!
subroutine getnp4old_ss(ex,crho,sigma,R, X, Y, Z, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz, &
drhodxx,drhodxy,drhodxz,drhodyy,drhodyz,drhodzz, &
dsigmadxx,dsigmadxy,dsigmadxz,dsigmadyy,dsigmadyz,dsigmadzz, &
dRdxx,dRdxy,dRdxz,dRdyy,dRdyz,dRdzz, &
chi, trK, &
dxx, gxy, gxz, dyy, gyz, dzz, &
Axx, Axy, Axz, Ayy, Ayz, Azz, &
Gmx, Gmy, Gmz, &
Lap, Sfx, Sfy, Sfz,Rpsi4,Ipsi4, symmetry,sst)
implicit none
!~~~~~~> Input parameters:
integer,intent(in ):: ex(1:3),symmetry,sst
double precision,intent(in),dimension(ex(1))::crho
double precision,intent(in),dimension(ex(2))::sigma
double precision,intent(in),dimension(ex(3))::R
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: X,Y,Z
double precision,intent(in),dimension(ex(1),ex(2),ex(3))::drhodx, drhody, drhodz
double precision,intent(in),dimension(ex(1),ex(2),ex(3))::dsigmadx,dsigmady,dsigmadz
double precision,intent(in),dimension(ex(1),ex(2),ex(3))::dRdx,dRdy,dRdz
double precision,intent(in),dimension(ex(1),ex(2),ex(3))::drhodxx,drhodxy,drhodxz,drhodyy,drhodyz,drhodzz
double precision,intent(in),dimension(ex(1),ex(2),ex(3))::dsigmadxx,dsigmadxy,dsigmadxz,dsigmadyy,dsigmadyz,dsigmadzz
double precision,intent(in),dimension(ex(1),ex(2),ex(3))::dRdxx,dRdxy,dRdxz,dRdyy,dRdyz,dRdzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: chi,trK
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: dxx,gxy,gxz,dyy,gyz,dzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Axx,Axy,Axz,Ayy,Ayz,Azz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Gmx,Gmy,Gmz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Lap,Sfx,Sfy,Sfz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: Rpsi4,Ipsi4
!~~~~~~> Other variables:
real*8, dimension(ex(1),ex(2),ex(3)) :: gupxx,gupxy,gupxz
real*8, dimension(ex(1),ex(2),ex(3)) :: gupyy,gupyz,gupzz
real*8, dimension(ex(1),ex(2),ex(3)) :: ep4phi,alpn1
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gyy,gzz
real*8, dimension(ex(1),ex(2),ex(3)) :: phi,phix,phiy,phiz
real*8, dimension(ex(1),ex(2),ex(3)) :: phixx,phixy,phixz,phiyy,phiyz,phizz
real*8, dimension(ex(1),ex(2),ex(3)) :: tRxyxy, tRxyxz, tRxyyz, tRxzxz, tRxzyz, tRyzyz
real*8, dimension(ex(1),ex(2),ex(3)) :: Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz
real*8, dimension(ex(1),ex(2),ex(3)) :: vx,vy,vz,ux,uy,uz,wx,wy,wz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamxxx, Gamxxy, Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamxyy, Gamxyz, Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamyxx, Gamyxy, Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamyyy, Gamyyz, Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamzxx, Gamzxy, Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)) :: Gamzyy, Gamzyz, Gamzzz
real*8, dimension(ex(1),ex(2),ex(3)) :: tRxx,tRxy,tRxz,tRyy,tRyz,tRzz
real*8, dimension(ex(1),ex(2),ex(3)) :: Rxx, Rxy, Rxz, Ryy, Ryz, Rzz
real*8, dimension(ex(1),ex(2),ex(3)) :: Kxx,Kxy,Kxz,Kyy,Kyz,Kzz
!D_i K_jk ---> DKijk
real*8, dimension(ex(1),ex(2),ex(3)) :: DKxxx,DKxxy,DKxxz,DKxyy,DKxyz,DKxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: DKyxx,DKyxy,DKyxz,DKyyy,DKyyz,DKyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: DKzxx,DKzxy,DKzxz,DKzyy,DKzyz,DKzzz
! Aij,k --> stored as Aijk
real*8, dimension(ex(1),ex(2),ex(3))::Axxx,Axxy,Axxz
real*8, dimension(ex(1),ex(2),ex(3))::Axyx,Axyy,Axyz
real*8, dimension(ex(1),ex(2),ex(3))::Axzx,Axzy,Axzz
real*8, dimension(ex(1),ex(2),ex(3))::Ayyx,Ayyy,Ayyz
real*8, dimension(ex(1),ex(2),ex(3))::Ayzx,Ayzy,Ayzz
real*8, dimension(ex(1),ex(2),ex(3))::Azzx,Azzy,Azzz
! trK,i
real*8, dimension(ex(1),ex(2),ex(3))::Kx,Ky,Kz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamxxx,ass_Gamxxy,ass_Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamxyy,ass_Gamxyz,ass_Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamyxx,ass_Gamyxy,ass_Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamyyy,ass_Gamyyz,ass_Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamzxx,ass_Gamzxy,ass_Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)) :: ass_Gamzyy,ass_Gamzyz,ass_Gamzzz
! first order partial derivative of metric
real*8, dimension(ex(1),ex(2),ex(3)) :: gxxx,gxxy,gxxz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxyx,gxyy,gxyz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxzx,gxzy,gxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gyyx,gyyy,gyyz
real*8, dimension(ex(1),ex(2),ex(3)) :: gyzx,gyzy,gyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gzzx,gzzy,gzzz
! second order partial derivative of metric
real*8, dimension(ex(1),ex(2),ex(3)) :: gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz
real*8, parameter :: F1o4=2.5d-1,ONE=1.d0,TWO=2.d0,FOUR=4.d0
real*8, parameter :: SYM = 1.D0, ANTI= - 1.D0
real*8,parameter::TINYRR=1.d-14
phi = -0.25d0*dlog(chi+ONE)
!~~~~~~
gxx = dxx + ONE
gyy = dyy + ONE
gzz = dzz + ONE
! invert tilted metric
gupzz = gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
gupxx = ( gyy * gzz - gyz * gyz ) / gupzz
gupxy = - ( gxy * gzz - gyz * gxz ) / gupzz
gupxz = ( gxy * gyz - gyy * gxz ) / gupzz
gupyy = ( gxx * gzz - gxz * gxz ) / gupzz
gupyz = - ( gxx * gyz - gxy * gxz ) / gupzz
gupzz = ( gxx * gyy - gxy * gxy ) / gupzz
alpn1 = Lap + ONE
ep4phi = dexp( FOUR * Phi )
!~~~~~~>
call fderivs_shc(ex,dxx,gxxx,gxxy,gxxz,crho,sigma,R, SYM, SYM,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,gxy,gxyx,gxyy,gxyz,crho,sigma,R,ANTI,ANTI,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,gxz,gxzx,gxzy,gxzz,crho,sigma,R,ANTI,SYM ,ANTI,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,dyy,gyyx,gyyy,gyyz,crho,sigma,R, SYM, SYM,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,gyz,gyzx,gyzy,gyzz,crho,sigma,R,SYM ,ANTI,ANTI,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,dzz,gzzx,gzzy,gzzz,crho,sigma,R, SYM, SYM,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fdderivs_shc(ex,dxx,gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz,crho,sigma,R, SYM, SYM,SYM ,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz, &
drhodxx,drhodxy,drhodxz,drhodyy,drhodyz,drhodzz, &
dsigmadxx,dsigmadxy,dsigmadxz,dsigmadyy,dsigmadyz,dsigmadzz, &
dRdxx,dRdxy,dRdxz,dRdyy,dRdyz,dRdzz)
call fdderivs_shc(ex,dyy,gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz,crho,sigma,R, SYM, SYM,SYM ,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz, &
drhodxx,drhodxy,drhodxz,drhodyy,drhodyz,drhodzz, &
dsigmadxx,dsigmadxy,dsigmadxz,dsigmadyy,dsigmadyz,dsigmadzz, &
dRdxx,dRdxy,dRdxz,dRdyy,dRdyz,dRdzz)
call fdderivs_shc(ex,dzz,gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz,crho,sigma,R, SYM, SYM,SYM ,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz, &
drhodxx,drhodxy,drhodxz,drhodyy,drhodyz,drhodzz, &
dsigmadxx,dsigmadxy,dsigmadxz,dsigmadyy,dsigmadyz,dsigmadzz, &
dRdxx,dRdxy,dRdxz,dRdyy,dRdyz,dRdzz)
call fdderivs_shc(ex,gxy,gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz,crho,sigma,R,ANTI,ANTI,SYM ,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz, &
drhodxx,drhodxy,drhodxz,drhodyy,drhodyz,drhodzz, &
dsigmadxx,dsigmadxy,dsigmadxz,dsigmadyy,dsigmadyz,dsigmadzz, &
dRdxx,dRdxy,dRdxz,dRdyy,dRdyz,dRdzz)
call fdderivs_shc(ex,gxz,gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz,crho,sigma,R,ANTI,SYM ,ANTI,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz, &
drhodxx,drhodxy,drhodxz,drhodyy,drhodyz,drhodzz, &
dsigmadxx,dsigmadxy,dsigmadxz,dsigmadyy,dsigmadyz,dsigmadzz, &
dRdxx,dRdxy,dRdxz,dRdyy,dRdyz,dRdzz)
call fdderivs_shc(ex,gyz,gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz,crho,sigma,R,SYM ,ANTI,ANTI,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz, &
drhodxx,drhodxy,drhodxz,drhodyy,drhodyz,drhodzz, &
dsigmadxx,dsigmadxy,dsigmadxz,dsigmadyy,dsigmadyz,dsigmadzz, &
dRdxx,dRdxy,dRdxz,dRdyy,dRdyz,dRdzz)
call kind1_connection(ex, gxxx,gxyx,gxzx,gyyx,gyzx,gzzx, &
gxxy,gxyy,gxzy,gyyy,gyzy,gzzy, &
gxxz,gxyz,gxzz,gyyz,gyzz,gzzz, &
ass_Gamxxx, ass_Gamxxy, ass_Gamxxz,&
ass_Gamxyy, ass_Gamxyz, ass_Gamxzz,&
ass_Gamyxx, ass_Gamyxy, ass_Gamyxz,&
ass_Gamyyy, ass_Gamyyz, ass_Gamyzz,&
ass_Gamzxx, ass_Gamzxy, ass_Gamzxz,&
ass_Gamzyy, ass_Gamzyz, ass_Gamzzz)
call kind2_connection(ex, gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
ass_Gamxxx, ass_Gamxxy, ass_Gamxxz, &
ass_Gamxyy, ass_Gamxyz, ass_Gamxzz, &
ass_Gamyxx, ass_Gamyxy, ass_Gamyxz, &
ass_Gamyyy, ass_Gamyyz, ass_Gamyzz, &
ass_Gamzxx, ass_Gamzxy, ass_Gamzxz, &
ass_Gamzyy, ass_Gamzyz, ass_Gamzzz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz)
!~~~~~~> derivs of conformal factor
call fderivs_shc(ex,phi,phix,phiy,phiz,crho,sigma,R, SYM, SYM,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fdderivs_shc(ex,phi,phixx,phixy,phixz,phiyy,phiyz,phizz,crho,sigma,R,SYM ,SYM ,SYM ,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz, &
drhodxx,drhodxy,drhodxz,drhodyy,drhodyz,drhodzz, &
dsigmadxx,dsigmadxy,dsigmadxz,dsigmadyy,dsigmadyz,dsigmadzz, &
dRdxx,dRdxy,dRdxz,dRdyy,dRdyz,dRdzz)
call xcov_deriv(ex, phix, phiy, phiz, &
phixx, phixy, phixz, phiyy, phiyz, phizz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz)
!~~~~~~> get spatial Riemann curvature
call adm_riemann(ex, gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz, &
gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz, &
gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz, &
gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz, &
gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz, &
gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz, &
ass_Gamxxx,ass_Gamxxy,ass_Gamxxz, &
ass_Gamxyy,ass_Gamxyz,ass_Gamxzz, &
ass_Gamyxx,ass_Gamyxy,ass_Gamyxz, &
ass_Gamyyy,ass_Gamyyz,ass_Gamyzz, &
ass_Gamzxx,ass_Gamzxy,ass_Gamzxz, &
ass_Gamzyy,ass_Gamzyz,ass_Gamzzz, &
tRxyxy, tRxyxz, tRxyyz, tRxzxz, tRxzyz, tRyzyz)
call get_physical_riemann(ex, ep4phi, &
dxx , gxy , gxz , dyy , gyz , dzz , &
gupxx , gupxy , gupxz , gupyy , gupyz , gupzz , &
phix , phiy , phiz , &
phixx , phixy , phixz , phiyy , phiyz , phizz , &
tRxyxy, tRxyxz, tRxyyz, tRxzxz, tRxzyz, tRyzyz, &
Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz)
!~~~~~~> get spatial Ricci tensor
call adm_ricci(ex, gupxx, gupxy, gupxz, gupyy, gupyz, gupzz, &
tRxyxy,tRxyxz,tRxyyz,tRxzxz,tRxzyz,tRyzyz, &
tRxx, tRxy, tRxz, tRyy, tRyz, tRzz)
call get_physical_ricci(ex,dxx,gxy,gxz,dyy,gyz,dzz,phix,phiy,phiz, &
phixx,phixy,phixz,phiyy,phiyz,phizz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
tRxx, tRxy, tRxz, tRyy, tRyz, tRzz, &
Rxx, Rxy, Rxz, Ryy, Ryz, Rzz)
!~~~~~~> get the real spatial extrinsic curvature
call get_physical_k(ex, phi, trK, dxx, gxy, gxz, dyy, gyz, dzz, &
Axx, Axy, Axz, Ayy, Ayz, Azz, &
Kxx, Kxy, Kxz, Kyy, Kyz, Kzz)
!~~~~~~> derivs of trace of extrinsic curvature
call fderivs_shc(ex,trK,Kx,Ky,Kz,crho,sigma,R, SYM, SYM,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
!~~~~~~> derivs of tilde extrinsic curvature
call fderivs_shc(ex,Axx,Axxx,Axxy,Axxz,crho,sigma,R, SYM, SYM,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,Axy,Axyx,Axyy,Axyz,crho,sigma,R,ANTI,ANTI,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,Axz,Axzx,Axzy,Axzz,crho,sigma,R,ANTI,SYM ,ANTI,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,Ayy,Ayyx,Ayyy,Ayyz,crho,sigma,R, SYM, SYM,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,Ayz,Ayzx,Ayzy,Ayzz,crho,sigma,R,SYM ,ANTI,ANTI,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
call fderivs_shc(ex,Azz,Azzx,Azzy,Azzz,crho,sigma,R, SYM, SYM,SYM,Symmetry,0,sst, &
drhodx, drhody, drhodz, &
dsigmadx,dsigmady,dsigmadz, &
dRdx,dRdy,dRdz)
!~~~~~~> derivs of extrinsic curvature, Kij
call get_diff_physical_k(ex, phi, trK, Kx, Ky, Kz, phix, phiy, phiz, &
dxx, gxy, gxz, dyy, gyz, dzz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
Axx, Axy, Axz, Ayy, Ayz, Azz, &
Axxx, Axxy, Axxz, Axyx, Axyy, Axyz, &
Axzx, Axzy, Axzz, Ayyx, Ayyy, Ayyz, &
Ayzx, Ayzy, Ayzz, Azzx, Azzy, Azzz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz, &
Kxx, Kxy, Kxz, Kyy, Kyz, Kzz, &
DKxxx, DKxxy, DKxxz, DKxyy, DKxyz, DKxzz, &
DKyxx, DKyxy, DKyxz, DKyyy, DKyyz, DKyzz, &
DKzxx, DKzxy, DKzxz, DKzyy, DKzyz, DKzzz)
!~~~~~~> get the Gram-Schmidt orthonormalize triad coordinate
#if (tetradtype == 0)
call get_triad0_ss(ex, X, Y, Z, ep4phi, gxx, gxy, gxz, gyy, gyz, gzz, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
#elif (tetradtype == 1)
call get_triad1_ss(ex, X, Y, Z, ep4phi, gxx, gxy, gxz, gyy, gyz, gzz, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
#elif (tetradtype == 2)
call get_triad2_ss(ex, X, Y, Z, ep4phi, gxx, gxy, gxz, gyy, gyz, gzz, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
#endif
!~~~~~~> compute the Newnamm-Penrose psi4 which split real and image part
ep4phi = ONE / ep4phi
call bssn_compute_psi4(ex,ep4phi, alpn1, Sfx, Sfy, Sfz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
vx,vy,vz,ux,uy,uz,wx,wy,wz, &
trK,Kxx,Kxy,Kxz,Kyy,Kyz,Kzz, &
Rxyxy,Rxyxz,Rxyyz,Rxzxz,Rxzyz,Ryzyz, &
Rxx, Rxy, Rxz, Ryy, Ryz, Rzz, &
DKxxx,DKxxy,DKxxz,DKxyy,DKxyz,DKxzz, &
DKyxx,DKyxy,DKyxz,DKyyy,DKyyz,DKyzz, &
DKzxx,DKzxy,DKzxz,DKzyy,DKzyz,DKzzz, Rpsi4, Ipsi4)
return
end subroutine getnp4old_ss
!----------------------------------------------------------!
! !
! derivatives related to 3-dimensional Riemann slice !
! !
!----------------------------------------------------------!
!-----------------------------------------------------------------------------
! Interface to compute the first order derivative of metric
!-----------------------------------------------------------------------------
subroutine d1metric(ex,X,Y,Z, &
dxx ,gxy ,gxz ,dyy ,gyz ,dzz , &
gxxx,gxyx,gxzx,gyyx,gyzx,gzzx, &
gxxy,gxyy,gxzy,gyyy,gyzy,gzzy, &
gxxz,gxyz,gxzz,gyyz,gyzz,gzzz, symmetry)
implicit none
!~~~~~~ Input parameters:
integer, intent(in ) :: ex(1:3),symmetry
real*8, intent(in ):: X(1:ex(1)),Y(1:ex(2)),Z(1:ex(3))
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: dxx,gxy,gxz,dyy,gyz,dzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: gxxx,gxyx,gxzx,gyyx,gyzx,gzzx
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: gxxy,gxyy,gxzy,gyyy,gyzy,gzzy
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: gxxz,gxyz,gxzz,gyyz,gyzz,gzzz
!~~~~~~ local variables
real*8, parameter :: SYM = 1.D0, ANTI= - 1.D0
!~~~~~~ 1st derivs of matric
call fderivs(ex,dxx,gxxx,gxxy,gxxz,X,Y,Z,SYM ,SYM ,SYM ,symmetry,0)
call fderivs(ex,gxy,gxyx,gxyy,gxyz,X,Y,Z,ANTI,ANTI,SYM ,symmetry,0)
call fderivs(ex,gxz,gxzx,gxzy,gxzz,X,Y,Z,ANTI,SYM ,ANTI,symmetry,0)
call fderivs(ex,dyy,gyyx,gyyy,gyyz,X,Y,Z,SYM ,SYM ,SYM ,symmetry,0)
call fderivs(ex,gyz,gyzx,gyzy,gyzz,X,Y,Z,SYM ,ANTI,ANTI,symmetry,0)
call fderivs(ex,dzz,gzzx,gzzy,gzzz,X,Y,Z,SYM ,SYM ,SYM ,symmetry,0)
return
end subroutine d1metric
!-----------------------------------------------------------------------------
! Interface to compute the second order derivative of metric
!-----------------------------------------------------------------------------
subroutine d2metric(ex,X,Y,Z, &
dxx, gxy, gxz, dyy, gyz, dzz, &
gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz, &
gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz, &
gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz, &
gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz, &
gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz, &
gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz, symmetry)
implicit none
!~~~~~~ Input parameters:
integer, intent(in ) :: ex(1:3),symmetry
real*8, intent(in ):: X(1:ex(1)),Y(1:ex(2)),Z(1:ex(3))
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: dxx,gxy,gxz,dyy,gyz,dzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz
!~~~~~~ local variables
real*8, parameter :: SYM = 1.D0, ANTI= - 1.D0
!~~~~~~ 2nd derivs of matric
call fdderivs(ex,dxx,gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz,X,Y,Z, &
SYM ,SYM ,SYM ,symmetry,0)
call fdderivs(ex,gxy,gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz,X,Y,Z, &
ANTI,ANTI,SYM ,symmetry,0)
call fdderivs(ex,gxz,gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz,X,Y,Z, &
ANTI,SYM ,ANTI,symmetry,0)
call fdderivs(ex,dyy,gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz,X,Y,Z, &
SYM, SYM ,SYM ,symmetry,0)
call fdderivs(ex,gyz,gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz,X,Y,Z, &
SYM ,ANTI,ANTI,symmetry,0)
call fdderivs(ex,dzz,gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz,X,Y,Z, &
SYM ,SYM ,SYM ,symmetry,0)
return
end subroutine d2metric
!----------------------------------------------------------!
! !
! algebraic computation based on geometric quantites !
! and their partial derivatives related to 3-dimensional !
! Riemann slice !
! !
!----------------------------------------------------------!
!-----------------------------------------------------------------------------
! Get first kind of connection coefficients
! based on first order derivative of metric
! ass_Gam_ijk = 1/2 *(g_ij,k + g_ki,j - g_jk,i)
!-----------------------------------------------------------------------------
subroutine kind1_connection(ex,gxxx,gxyx,gxzx,gyyx,gyzx,gzzx, &
gxxy,gxyy,gxzy,gyyy,gyzy,gzzy, &
gxxz,gxyz,gxzz,gyyz,gyzz,gzzz, &
ass_Gamxxx, ass_Gamxxy, ass_Gamxxz, &
ass_Gamxyy, ass_Gamxyz, ass_Gamxzz, &
ass_Gamyxx, ass_Gamyxy, ass_Gamyxz, &
ass_Gamyyy, ass_Gamyyz, ass_Gamyzz, &
ass_Gamzxx, ass_Gamzxy, ass_Gamzxz, &
ass_Gamzyy, ass_Gamzyz, ass_Gamzzz)
implicit none
!~~~~~~> Input parameters:
integer, intent(in) :: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gxxx,gxyx,gxzx
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gyyx,gyzx,gzzx
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gxxy,gxyy,gxzy
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gyyy,gyzy,gzzy
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gxxz,gxyz,gxzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gyyz,gyzz,gzzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: ass_Gamxxx,ass_Gamxxy,ass_Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: ass_Gamxyy,ass_Gamxyz,ass_Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: ass_Gamyxx,ass_Gamyxy,ass_Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: ass_Gamyyy,ass_Gamyyz,ass_Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: ass_Gamzxx,ass_Gamzxy,ass_Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: ass_Gamzyy,ass_Gamzyz,ass_Gamzzz
!~~~~~~> Other variables:
real*8, parameter :: HLF=0.5d0
!~~~~~~= Get Connection coefficients
! ass_Gam_ijk = 1/2 *(g_ij,k + g_ki,j - g_jk,i)
ass_Gamxxx = HLF * ( gxxx )
ass_Gamyxx = HLF * ( gxyx + gxyx - gxxy )
ass_Gamzxx = HLF * ( gxzx + gxzx - gxxz )
ass_Gamxyy = HLF * ( gxyy + gxyy - gyyx )
ass_Gamyyy = HLF * ( gyyy )
ass_Gamzyy = HLF * ( gyzy + gyzy - gyyz )
ass_Gamxzz = HLF * ( gxzz + gxzz - gzzx )
ass_Gamyzz = HLF * ( gyzz + gyzz - gzzy )
ass_Gamzzz = HLF * ( gzzz )
ass_Gamxxy = HLF * ( gxxy + gxyx - gxyx )
ass_Gamyxy = HLF * ( gxyy + gyyx - gxyy )
ass_Gamzxy = HLF * ( gxzy + gyzx - gxyz )
ass_Gamxxz = HLF * ( gxxz + gxzx - gxzx )
ass_Gamyxz = HLF * ( gxyz + gyzx - gxzy )
ass_Gamzxz = HLF * ( gxzz + gzzx - gxzz )
ass_Gamxyz = HLF * ( gxyz + gxzy - gyzx )
ass_Gamyyz = HLF * ( gyyz + gyzy - gyzy )
ass_Gamzyz = HLF * ( gyzz + gzzy - gyzz )
return
end subroutine kind1_connection
!-----------------------------------------------------------------------------
! Get second kind of connection coefficients
! based on first kind of connection coefficients
! and gup
!-----------------------------------------------------------------------------
subroutine kind2_connection(ex,gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
ass_Gamxxx, ass_Gamxxy, ass_Gamxxz, &
ass_Gamxyy, ass_Gamxyz, ass_Gamxzz, &
ass_Gamyxx, ass_Gamyxy, ass_Gamyxz, &
ass_Gamyyy, ass_Gamyyz, ass_Gamyzz, &
ass_Gamzxx, ass_Gamzxy, ass_Gamzxz, &
ass_Gamzyy, ass_Gamzyz, ass_Gamzzz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz)
implicit none
!~~~~~~> Input parameters:
integer, intent(in) :: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: gupxx,gupxy,gupxz,gupyy,gupyz,gupzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: ass_Gamxxx,ass_Gamxxy,ass_Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: ass_Gamxyy,ass_Gamxyz,ass_Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: ass_Gamyxx,ass_Gamyxy,ass_Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: ass_Gamyyy,ass_Gamyyz,ass_Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: ass_Gamzxx,ass_Gamzxy,ass_Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: ass_Gamzyy,ass_Gamzyz,ass_Gamzzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: Gamxxx, Gamxxy, Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: Gamxyy, Gamxyz, Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: Gamyxx, Gamyxy, Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: Gamyyy, Gamyyz, Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: Gamzxx, Gamzxy, Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: Gamzyy, Gamzyz, Gamzzz
!~~~~~~> Other variables:
Gamxxx = gupxx * ass_Gamxxx + gupxy * ass_Gamyxx + gupxz * ass_Gamzxx
Gamxxy = gupxx * ass_Gamxxy + gupxy * ass_Gamyxy + gupxz * ass_Gamzxy
Gamxxz = gupxx * ass_Gamxxz + gupxy * ass_Gamyxz + gupxz * ass_Gamzxz
Gamxyy = gupxx * ass_Gamxyy + gupxy * ass_Gamyyy + gupxz * ass_Gamzyy
Gamxyz = gupxx * ass_Gamxyz + gupxy * ass_Gamyyz + gupxz * ass_Gamzyz
Gamxzz = gupxx * ass_Gamxzz + gupxy * ass_Gamyzz + gupxz * ass_Gamzzz
Gamyxx = gupxy * ass_Gamxxx + gupyy * ass_Gamyxx + gupyz * ass_Gamzxx
Gamyxy = gupxy * ass_Gamxxy + gupyy * ass_Gamyxy + gupyz * ass_Gamzxy
Gamyxz = gupxy * ass_Gamxxz + gupyy * ass_Gamyxz + gupyz * ass_Gamzxz
Gamyyy = gupxy * ass_Gamxyy + gupyy * ass_Gamyyy + gupyz * ass_Gamzyy
Gamyyz = gupxy * ass_Gamxyz + gupyy * ass_Gamyyz + gupyz * ass_Gamzyz
Gamyzz = gupxy * ass_Gamxzz + gupyy * ass_Gamyzz + gupyz * ass_Gamzzz
Gamzxx = gupxz * ass_Gamxxx + gupyz * ass_Gamyxx + gupzz * ass_Gamzxx
Gamzxy = gupxz * ass_Gamxxy + gupyz * ass_Gamyxy + gupzz * ass_Gamzxy
Gamzxz = gupxz * ass_Gamxxz + gupyz * ass_Gamyxz + gupzz * ass_Gamzxz
Gamzyy = gupxz * ass_Gamxyy + gupyz * ass_Gamyyy + gupzz * ass_Gamzyy
Gamzyz = gupxz * ass_Gamxyz + gupyz * ass_Gamyyz + gupzz * ass_Gamzyz
Gamzzz = gupxz * ass_Gamxzz + gupyz * ass_Gamyzz + gupzz * ass_Gamzzz
return
end subroutine kind2_connection
!----------------------------------------------------------------------
! compute Riemann tensor for three dimensional space
! based on second derivatives of metric
! and first knid and second kind of connection
!----------------------------------------------------------------------
subroutine adm_riemann(ex,gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz, &
gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz, &
gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz, &
gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz, &
gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz, &
gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz, &
ass_Gamxxx,ass_Gamxxy,ass_Gamxxz, &
ass_Gamxyy,ass_Gamxyz,ass_Gamxzz, &
ass_Gamyxx,ass_Gamyxy,ass_Gamyxz, &
ass_Gamyyy,ass_Gamyyz,ass_Gamyzz, &
ass_Gamzxx,ass_Gamzxy,ass_Gamzxz, &
ass_Gamzyy,ass_Gamzyz,ass_Gamzzz, &
Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz)
implicit none
!~~~~~~ argument variables
integer,intent(in ) :: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gxxxx,gxxxy,gxxxz,gxxyy,gxxyz,gxxzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gxyxx,gxyxy,gxyxz,gxyyy,gxyyz,gxyzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gxzxx,gxzxy,gxzxz,gxzyy,gxzyz,gxzzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gyyxx,gyyxy,gyyxz,gyyyy,gyyyz,gyyzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gyzxx,gyzxy,gyzxz,gyzyy,gyzyz,gyzzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gzzxx,gzzxy,gzzxz,gzzyy,gzzyz,gzzzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Gamxxx, Gamxxy, Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Gamxyy, Gamxyz, Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Gamyxx, Gamyxy, Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Gamyyy, Gamyyz, Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Gamzxx, Gamzxy, Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Gamzyy, Gamzyz, Gamzzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ass_Gamxxx,ass_Gamxxy,ass_Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ass_Gamxyy,ass_Gamxyz,ass_Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ass_Gamyxx,ass_Gamyxy,ass_Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ass_Gamyyy,ass_Gamyyz,ass_Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ass_Gamzxx,ass_Gamzxy,ass_Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ass_Gamzyy,ass_Gamzyz,ass_Gamzzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz
!~~~~~~local variables
real*8, parameter :: HLF=0.5d0
!R_ijkl = HLF *(@_jk g_il + @_il g_jk - @_jl g_ik - @_ik g_jl)
! + Gam_rjk Gam^r_il - Gam_rjl Gam^r_ik
Rxyxy = HLF *( gxyxy + gxyxy - gxxyy - gyyxx ) + &
(ass_Gamxxy * Gamxxy + ass_Gamyxy * Gamyxy + ass_Gamzxy * Gamzxy) - &
(ass_Gamxyy * Gamxxx + ass_Gamyyy * Gamyxx + ass_Gamzyy * Gamzxx)
Rxyxz = HLF *( gxzxy + gxyxz - gxxyz - gyzxx ) + &
(ass_Gamxxy * Gamxxz + ass_Gamyxy * Gamyxz + ass_Gamzxy * Gamzxz) - &
(ass_Gamxyz * Gamxxx + ass_Gamyyz * Gamyxx + ass_Gamzyz * Gamzxx)
Rxyyz = HLF *( gxzyy + gyyxz - gxyyz - gyzxy ) + &
(ass_Gamxyy * Gamxxz + ass_Gamyyy * Gamyxz + ass_Gamzyy * Gamzxz) - &
(ass_Gamxyz * Gamxxy + ass_Gamyyz * Gamyxy + ass_Gamzyz * Gamzxy)
Rxzxz = HLF *( gxzxz + gxzxz - gxxzz - gzzxx ) + &
(ass_Gamxxz * Gamxxz + ass_Gamyxz * Gamyxz + ass_Gamzxz * Gamzxz) - &
(ass_Gamxzz * Gamxxx + ass_Gamyzz * Gamyxx + ass_Gamzzz * Gamzxx)
Rxzyz = HLF *( gxzyz + gyzxz - gxyzz - gzzxy ) + &
(ass_Gamxyz * Gamxxz + ass_Gamyyz * Gamyxz + ass_Gamzyz * Gamzxz) - &
(ass_Gamxzz * Gamxxy + ass_Gamyzz * Gamyxy + ass_Gamzzz * Gamzxy)
Ryzyz = HLF *( gyzyz + gyzyz - gyyzz - gzzyy ) + &
(ass_Gamxyz * Gamxyz + ass_Gamyyz * Gamyyz + ass_Gamzyz * Gamzyz) - &
(ass_Gamxzz * Gamxyy + ass_Gamyzz * Gamyyy + ass_Gamzzz * Gamzyy)
return
end subroutine adm_riemann
!-----------------------------------------------------------------------------
! Get Ricci tensor of metric g from Riemann tensor
! for adm form
! R_ij = gup^kl * R_ikjl
!-----------------------------------------------------------------------------
subroutine adm_ricci(ex, gupxx, gupxy, gupxz, gupyy, gupyz, gupzz, &
Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz, &
Rxx, Rxy, Rxz, Ryy, Ryz, Rzz)
implicit none
!~~~~~~> Input parameters:
integer, intent(in ) :: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gupxx,gupxy,gupxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gupyy,gupyz,gupzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Rxyxy, Rxyxz, Rxyyz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Rxzxz, Rxzyz, Ryzyz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: Rxx,Rxy,Rxz,Ryy,Ryz,Rzz
Rxx = gupyy * Rxyxy + gupyz * Rxyxz + gupyz * Rxyxz + gupzz * Rxzxz
Rxy = - gupxy * Rxyxy + gupyz * Rxyyz - gupxz * Rxyxz + gupzz * Rxzyz
Rxz = - gupxy * Rxyxz - gupyy * Rxyyz - gupxz * Rxzxz - gupyz * Rxzyz
Ryy = gupxx * Rxyxy - gupxz * Rxyyz - gupxz * Rxyyz + gupzz * Ryzyz
Ryz = gupxx * Rxyxz + gupxy * Rxyyz - gupxz * Rxzyz - gupyz * Ryzyz
Rzz = gupxx * Rxzxz + gupxy * Rxzyz + gupxy * Rxzyz + gupyy * Ryzyz
return
end subroutine adm_ricci
!-----------------------------------------------------------------------------
! raise index
!-----------------------------------------------------------------------------
subroutine raise(ex,fx,fy,fz,fupx,fupy,fupz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz)
implicit none
!~~~~~~ Input parameters:
integer, intent(in ) :: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: fx,fy,fz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gupxx, gupxy, gupxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gupyy, gupyz, gupzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: fupx,fupy,fupz
fupx = gupxx * fx + gupxy * fy + gupxz * fz
fupy = gupxy * fx + gupyy * fy + gupyz * fz
fupz = gupxz * fx + gupyz * fy + gupzz * fz
return
end subroutine raise
!-----------------------------------------------------------------------------
! lower index
!-----------------------------------------------------------------------------
subroutine lower(ex,fx,fy,fz,Lfx,Lfy,Lfz,gxx,gxy,gxz,gyy,gyz,gzz)
implicit none
!~~~~~~ Input parameters:
integer, intent(in ) :: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: fx,fy,fz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gxx,gxy,gxz,gyy,gyz,gzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: Lfx,Lfy,Lfz
Lfx = gxx * fx + gxy * fy + gxz * fz
Lfy = gxy * fx + gyy * fy + gyz * fz
Lfz = gxz * fx + gyz * fy + gzz * fz
return
end subroutine lower
!----------------------------------------------------------------------------------
! inner product of two three dimensional vectors with metric g_ij
! metric here do not upto ONE
!----------------------------------------------------------------------------------
subroutine InnerProd(ex,norm,ux,uy,uz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
implicit none
!~~~~~~ argument variables
integer,intent(in ):: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)),intent(in)::ux,uy,uz,vx,vy,vz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in)::gxx,gxy,gxz,gyy,gyz,gzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out)::norm
norm = gxx * ux * vx + gxy * ux * vy + gxz * ux * vz &
+ gxy * uy * vx + gyy * uy * vy + gyz * uy * vz &
+ gxz * uz * vx + gyz * uz * vy + gzz * uz * vz
return
end subroutine InnerProd
!----------------------------------------------------------!
! !
! algebraic computation based on geometric quantites !
! and their partial derivatives related to 3-dimensional !
! Riemann slice !
! !
! * for BSSN form * !
!----------------------------------------------------------!
!-----------------------------------------------------------------------------
! second order covariant derivatives w.r.t. *untilded* (i.e. physical) metric
! of *symmetric* variable of scalar field
!-----------------------------------------------------------------------------
subroutine fnt_cov_s_dderiv(ex, fx, fy, fz, &
fxx, fxy, fxz, fyy, fyz, fzz, &
phix, phiy, phiz, &
dxx, gxy, gxz, dyy, gyz, dzz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
Gmxxx,Gmxxy,Gmxxz,Gmxyy,Gmxyz,Gmxzz, &
Gmyxx,Gmyxy,Gmyxz,Gmyyy,Gmyyz,Gmyzz, &
Gmzxx,Gmzxy,Gmzxz,Gmzyy,Gmzyz,Gmzzz)
implicit none
!~~~~~~ Input arguments
integer, intent(in ) :: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: fx,fy,fz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: phix,phiy,phiz
! tilted Christofel symble
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gmxxx, Gmxxy, Gmxxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gmxyy, Gmxyz, Gmxzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gmyxx, Gmyxy, Gmyxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gmyyy, Gmyyz, Gmyzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gmzxx, Gmzxy, Gmzxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gmzyy, Gmzyz, Gmzzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: dxx,gxy,gxz,dyy,gyz,dzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gupxx,gupxy,gupxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: gupyy,gupyz,gupzz
! input partial derivatives, output covariant derivative respect to physical metric
real*8, dimension(ex(1),ex(2),ex(3)), intent(inout) :: fxx,fxy,fxz,fyy,fyz,fzz
!~~~~~~ Other variables:
real*8, dimension(ex(1),ex(2),ex(3)) :: phiupx,phiupy,phiupz
real*8,parameter :: TWO = 2.d0
!~~~~~~ Make untilded Gamma's out of tilded ones - first raise index on phi_i...
phiupx = gupxx * phix + gupxy * phiy + gupxz * phiz
phiupy = gupxy * phix + gupyy * phiy + gupyz * phiz
phiupz = gupxz * phix + gupyz * phiy + gupzz * phiz
!~~~~~~ ... and then add reconstructed *untilded* Christofels...
fxx = fxx - ( Gmxxx + TWO * ( phix + phix - dxx * phiupx - phiupx ))* fx - &
( Gmyxx + TWO * ( - dxx * phiupy - phiupy ))* fy - &
( Gmzxx + TWO * ( - dxx * phiupz - phiupz ))* fz
fyy = fyy - ( Gmxyy + TWO * ( - dyy * phiupx - phiupx ))* fx - &
( Gmyyy + TWO * ( phiy + phiy - dyy * phiupy - phiupy ))* fy - &
( Gmzyy + TWO * ( - dyy * phiupz - phiupz ))* fz
fzz = fzz - ( Gmxzz + TWO * ( - dzz * phiupx - phiupx ))* fx - &
( Gmyzz + TWO * ( - dzz * phiupy - phiupy ))* fy - &
( Gmzzz + TWO * ( phiz + phiz - dzz * phiupz - phiupz ))* fz
fxy = fxy - ( Gmxxy + TWO * ( phiy - gxy * phiupx ))* fx - &
( Gmyxy + TWO * ( phix - gxy * phiupy ))* fy - &
( Gmzxy + TWO * ( - gxy * phiupz ))* fz
fxz = fxz - ( Gmxxz + TWO * ( phiz - gxz * phiupx ))* fx - &
( Gmyxz + TWO * ( - gxz * phiupy ))* fy - &
( Gmzxz + TWO * ( phix - gxz * phiupz ))* fz
fyz = fyz - ( Gmxyz + TWO * ( - gyz * phiupx ))* fx - &
( Gmyyz + TWO * ( phiz - gyz * phiupy ))* fy - &
( Gmzyz + TWO * ( phiy - gyz * phiupz ))* fz
return
end subroutine fnt_cov_s_dderiv
!-----------------------------------------------------------------------------
!
! Get physical riemann tensor
!
!-----------------------------------------------------------------------------
subroutine get_physical_riemann(ex, ep4phi, &
dxx, gxy, gxz, dyy, gyz, dzz, &
gupxx, gupxy, gupxz, gupyy, gupyz, gupzz, &
phix, phiy, phiz, &
phixx, phixy, phixz, phiyy, phiyz, phizz, &
tRxyxy, tRxyxz, tRxyyz, tRxzxz, tRxzyz, tRyzyz, &
Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz)
implicit none
!~~~~~~> Input parameters:
integer, intent(in ):: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: ep4phi
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: dxx,gxy,gxz,dyy,gyz,dzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: gupxx,gupxy,gupxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: gupyy,gupyz,gupzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: phix,phiy,phiz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: phixx,phixy,phixz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: phiyy,phiyz,phizz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: tRxyxy,tRxyxz,tRxyyz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: tRxzxz,tRxzyz,tRyzyz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out):: Rxyxy, Rxyxz, Rxyyz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out):: Rxzxz, Rxzyz, Ryzyz
!~~~~~~> Other variables:
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gyy,gzz
real*8, dimension(ex(1),ex(2),ex(3)) :: tmp
real*8,parameter::ONE = 1.d0, TWO = 2.d0, FOUR = 4.d0
gxx = dxx + ONE
gyy = dyy + ONE
gzz = dzz + ONE
!~~~~~~> R_ijkl = tilde R_ijkl + TWO *( gli * D_j D_k phi - glj * D_i D_k phi -
! gki * D_j D_l phi + gkj * D_i D_l phi )
! + FOUR*( gjl * D_i phi * D_k phi - gil * D_j phi * D_k phi -
! gjk * D_i phi * D_l phi + gik * D_j phi * D_l phi )
! + FOUR*( gjk * gil - gik * gjl )* g^mn * D_m phi * D_n phi
tmp = gupxx * phix * phix + gupyy * phiy * phiy + gupzz * phiz * phiz + &
TWO *( gupxy * phix * phiy + gupxz * phix * phiz + gupyz * phiy * phiz )
!~~~~~~> R_ijkl = tilde R_ijkl + TWO *( gli * phi_jk - glj * phi_ik -
! gki * phi_jl + gkj * phi_il )
! + FOUR*( gjl * phi_i * phi_k - gil * phi_j * phi_k -
! gjk * phi_i * phi_l + gik * phi_j * phi_l )
! + FOUR*( gjk * gil - gik * gjl )* tmp
Rxyxy = tRxyxy + TWO *( gxy * phixy - gyy * phixx - gxx * phiyy + gxy * phixy ) &
+ FOUR*( gyy * phix * phix - gxy * phiy * phix - &
gxy * phix * phiy + gxx * phiy * phiy ) &
+ FOUR*( gxy * gxy - gxx * gyy )* tmp
Rxyxz = tRxyxz + TWO *( gxz * phixy - gyz * phixx - gxx * phiyz + gxy * phixz ) &
+ FOUR*( gyz * phix * phix - gxz * phiy * phix - &
gxy * phix * phiz + gxx * phiy * phiz ) &
+ FOUR*( gxy * gxz - gxx * gyz )* tmp
Rxyyz = tRxyyz + TWO *( gxz * phiyy - gyz * phixy - gxy * phiyz + gyy * phixz ) &
+ FOUR*( gyz * phix * phiy - gxz * phiy * phiy - &
gyy * phix * phiz + gxy * phiy * phiz ) &
+ FOUR*( gyy * gxz - gxy * gyz )* tmp
Rxzxz = tRxzxz + TWO *( gxz * phixz - gzz * phixx - gxx * phizz + gxz * phixz ) &
+ FOUR*( gzz * phix * phix - gxz * phiz * phix - &
gxz * phix * phiz + gxx * phiz * phiz ) &
+ FOUR*( gxz * gxz - gxx * gzz )* tmp
Rxzyz = tRxzyz + TWO *( gxz * phiyz - gzz * phixy - gxy * phizz + gyz * phixz ) &
+ FOUR*( gzz * phix * phiy - gxz * phiz * phiy - &
gyz * phix * phiz + gxy * phiz * phiz ) &
+ FOUR*( gyz * gxz - gxy * gzz )* tmp
Ryzyz = tRyzyz + TWO *( gyz * phiyz - gzz * phiyy - gyy * phizz + gyz * phiyz ) &
+ FOUR*( gzz * phiy * phiy - gyz * phiz * phiy - &
gyz * phiy * phiz + gyy * phiz * phiz ) &
+ FOUR*( gyz * gyz - gyy * gzz )* tmp
!multipli with factor exp( 4 * phi)
Rxyxy = Rxyxy * ep4phi
Rxyxz = Rxyxz * ep4phi
Rxyyz = Rxyyz * ep4phi
Rxzxz = Rxzxz * ep4phi
Rxzyz = Rxzyz * ep4phi
Ryzyz = Ryzyz * ep4phi
return
end subroutine get_physical_riemann
!-----------------------------------------------------------------------------
!
! Get physical Ricci tensor
!
!-----------------------------------------------------------------------------
subroutine get_physical_ricci(ex,dxx,gxy,gxz,dyy,gyz,dzz,phix,phiy,phiz, &
phixx,phixy,phixz,phiyy,phiyz,phizz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
tRxx, tRxy, tRxz, tRyy, tRyz, tRzz, &
Rxx, Rxy, Rxz, Ryy, Ryz, Rzz)
implicit none
!~~~~~~ argument variables
integer, intent(in) :: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)), intent(in):: dxx,gxy,gxz,dyy,gyz,dzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in):: phix,phiy,phiz
! covariant derivative respect to tilted metric
real*8, dimension(ex(1),ex(2),ex(3)), intent(in):: phixx,phixy,phixz,phiyy,phiyz,phizz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in):: gupxx,gupxy,gupxz,gupyy,gupyz,gupzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: tRxx,tRxy,tRxz,tRyy,tRyz,tRzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: Rxx, Rxy, Rxz, Ryy, Ryz, Rzz
!~~~~~~ local variables
real*8, dimension(ex(1),ex(2),ex(3)) :: tempf
real*8,parameter::TWO = 2.d0, FOUR = 4.d0
!~~~~~~
tempf = TWO * (gupxx * ( phixx + TWO * phix * phix ) + &
gupyy * ( phiyy + TWO * phiy * phiy ) + &
gupzz * ( phizz + TWO * phiz * phiz ) + &
TWO * gupxy * ( phixy + TWO * phix * phiy ) + &
TWO * gupxz * ( phixz + TWO * phix * phiz ) + &
TWO * gupyz * ( phiyz + TWO * phiy * phiz ) )
! Add phi part to Ricci tensor:
Rxx = tRxx - TWO * phixx + FOUR * phix * phix - dxx * tempf - tempf
Ryy = tRyy - TWO * phiyy + FOUR * phiy * phiy - dyy * tempf - tempf
Rzz = tRzz - TWO * phizz + FOUR * phiz * phiz - dzz * tempf - tempf
Rxy = tRxy - TWO * phixy + FOUR * phix * phiy - gxy * tempf
Rxz = tRxz - TWO * phixz + FOUR * phix * phiz - gxz * tempf
Ryz = tRyz - TWO * phiyz + FOUR * phiy * phiz - gyz * tempf
return
end subroutine get_physical_ricci
!-----------------------------------------------------------------------------
!
! compute physical extrinic curver:
! Kij = exp( 4 * phi ) ( tilde Aij + F1o3 * tilde gij * trK )
!
!-----------------------------------------------------------------------------
subroutine get_physical_k(ex, phi, trK, dxx, gxy, gxz, dyy, gyz, dzz, &
Axx, Axy, Axz, Ayy, Ayz, Azz, &
Kxx, Kxy, Kxz, Kyy, Kyz, Kzz)
implicit none
!~~~~~~> Input parameters:
integer,dimension(3) , intent(in) :: ex
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: phi, trK
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: dxx,gxy,gxz,dyy,gyz,dzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in) :: Axx,Axy,Axz,Ayy,Ayz,Azz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out):: Kxx,Kxy,Kxz,Kyy,Kyz,Kzz
!~~~~~~> Other variables:
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx, gyy, gzz
real*8, parameter :: F1o3 = 1.d0 / 3.d0, ONE = 1.d0, FOUR = 4.d0
!~~~~~~>
gxx = dxx + ONE
gyy = dyy + ONE
gzz = dzz + ONE
Kzz = exp( FOUR * phi )
!~~~~~~>
Kxx = ( Axx + F1o3 * gxx * trK )* Kzz
Kxy = ( Axy + F1o3 * gxy * trK )* Kzz
Kxz = ( Axz + F1o3 * gxz * trK )* Kzz
Kyy = ( Ayy + F1o3 * gyy * trK )* Kzz
Kyz = ( Ayz + F1o3 * gyz * trK )* Kzz
Kzz = ( Azz + F1o3 * gzz * trK )* Kzz
return
end subroutine get_physical_k
!-------------------------------------------------------------------------------------------------------
!
! compute covariant derivatives of extrinic curver
!
!D_i K_jk stored as DKijk
!
! DKijk = e^(4 phi) (A_jk,i - Gam^l_ij A_lk - Gam^l_ik A_jl + 1/3 g_jk trK,i)
! - 2 K_ik phi,j + 2 g_ij g^lm phi,m K_lk
! - 2 K_ij phi,k + 2 g_ik g^lm phi,m K_lj
!-------------------------------------------------------------------------------------------------------
subroutine get_diff_physical_k(ex, phi, trK, Kx, Ky, Kz, phix, phiy, phiz, &
dxx, gxy, gxz, dyy, gyz, dzz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
Axx, Axy, Axz, Ayy, Ayz, Azz, &
Axxx, Axxy, Axxz, Axyx, Axyy, Axyz, &
Axzx, Axzy, Axzz, Ayyx, Ayyy, Ayyz, &
Ayzx, Ayzy, Ayzz, Azzx, Azzy, Azzz, &
Gmxxx,Gmxxy,Gmxxz,Gmxyy,Gmxyz,Gmxzz, &
Gmyxx,Gmyxy,Gmyxz,Gmyyy,Gmyyz,Gmyzz, &
Gmzxx,Gmzxy,Gmzxz,Gmzyy,Gmzyz,Gmzzz, &
Kxx, Kxy, Kxz, Kyy, Kyz, Kzz, &
DKxxx,DKxxy,DKxxz,DKxyy,DKxyz,DKxzz, &
DKyxx,DKyxy,DKyxz,DKyyy,DKyyz,DKyzz, &
DKzxx,DKzxy,DKzxz,DKzyy,DKzyz,DKzzz)
implicit none
!~~~~~~> Input parameters:
integer,dimension(3), intent(in) :: ex
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: phi,trK
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Kx,Ky,Kz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: phix,phiy,phiz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: dxx,gxy,gxz,dyy,gyz,dzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: gupxx,gupxy,gupxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: gupyy,gupyz,gupzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Axx,Axy,Axz,Ayy,Ayz,Azz
! Aij,k --> stored as Aijk
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Axxx,Axxy,Axxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Axyx,Axyy,Axyz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Axzx,Axzy,Axzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Ayyx,Ayyy,Ayyz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Ayzx,Ayzy,Ayzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Azzx,Azzy,Azzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Gmxxx,Gmxxy,Gmxxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Gmxyy,Gmxyz,Gmxzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Gmyxx,Gmyxy,Gmyxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Gmyyy,Gmyyz,Gmyzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Gmzxx,Gmzxy,Gmzxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Gmzyy,Gmzyz,Gmzzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ):: Kxx,Kxy,Kxz,Kyy,Kyz,Kzz
! D_i K_jk --> stored as DKijk
real*8, dimension(ex(1),ex(2),ex(3)), intent(out):: DKxxx,DKxxy,DKxxz,DKxyy,DKxyz,DKxzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out):: DKyxx,DKyxy,DKyxz,DKyyy,DKyyz,DKyzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out):: DKzxx,DKzxy,DKzxz,DKzyy,DKzyz,DKzzz
!~~~~~~> Other variables:
real*8, dimension(ex(1),ex(2),ex(3)):: phiupx,phiupy,phiupz
real*8, dimension(ex(1),ex(2),ex(3)):: phiupKx,phiupKy,phiupKz
real*8, dimension(ex(1),ex(2),ex(3)):: e4phi
real*8, dimension(ex(1),ex(2),ex(3)):: gxx,gyy,gzz
real*8,parameter::ONE = 1.d0, TWO = 2.d0, FOUR = 4.d0
real*8,parameter::F1o3 = 1.d0/3.d0
real*8, parameter :: SYM = 1.D0, ANTI= - 1.D0
!~~~~~~> Input translation
gxx = dxx + ONE
gyy = dyy + ONE
gzz = dzz + ONE
e4phi = dexp(FOUR * phi)
!~~~~~~>
phiupx = gupxx * phix + gupxy * phiy + gupxz * phiz
phiupy = gupxy * phix + gupyy * phiy + gupyz * phiz
phiupz = gupxz * phix + gupyz * phiy + gupzz * phiz
phiupKx = phiupx * Kxx + phiupy * Kxy + phiupz * Kxz
phiupKy = phiupx * Kxy + phiupy * Kyy + phiupz * Kyz
phiupKz = phiupx * Kxz + phiupy * Kyz + phiupz * Kzz
!~~~~~~> tmp = - Gam^l_ij A_lk - Gam^l_ik A_jl
DKxxx = - Gmxxx * Axx - Gmyxx * Axy - Gmzxx * Axz &
- Gmxxx * Axx - Gmyxx * Axy - Gmzxx * Axz
DKxxy = - Gmxxx * Axy - Gmyxx * Ayy - Gmzxx * Ayz &
- Gmxxy * Axx - Gmyxy * Axy - Gmzxy * Axz
DKxxz = - Gmxxx * Axz - Gmyxx * Ayz - Gmzxx * Azz &
- Gmxxz * Axx - Gmyxz * Axy - Gmzxz * Axz
DKxyy = - Gmxxy * Axy - Gmyxy * Ayy - Gmzxy * Ayz &
- Gmxxy * Axy - Gmyxy * Ayy - Gmzxy * Ayz
DKxyz = - Gmxxy * Axz - Gmyxy * Ayz - Gmzxy * Azz &
- Gmxxz * Axy - Gmyxz * Ayy - Gmzxz * Ayz
DKxzz = - Gmxxz * Axz - Gmyxz * Ayz - Gmzxz * Azz &
- Gmxxz * Axz - Gmyxz * Ayz - Gmzxz * Azz
DKyxx = - Gmxxy * Axx - Gmyxy * Axy - Gmzxy * Axz &
- Gmxxy * Axx - Gmyxy * Axy - Gmzxy * Axz
DKyxy = - Gmxxy * Axy - Gmyxy * Ayy - Gmzxy * Ayz &
- Gmxyy * Axx - Gmyyy * Axy - Gmzyy * Axz
DKyxz = - Gmxxy * Axz - Gmyxy * Ayz - Gmzxy * Azz &
- Gmxyz * Axx - Gmyyz * Axy - Gmzyz * Axz
DKyyy = - Gmxyy * Axy - Gmyyy * Ayy - Gmzyy * Ayz &
- Gmxyy * Axy - Gmyyy * Ayy - Gmzyy * Ayz
DKyyz = - Gmxyy * Axz - Gmyyy * Ayz - Gmzyy * Azz &
- Gmxyz * Axy - Gmyyz * Ayy - Gmzyz * Ayz
DKyzz = - Gmxyz * Axz - Gmyyz * Ayz - Gmzyz * Azz &
- Gmxyz * Axz - Gmyyz * Ayz - Gmzyz * Azz
DKzxx = - Gmxxz * Axx - Gmyxz * Axy - Gmzxz * Axz &
- Gmxxz * Axx - Gmyxz * Axy - Gmzxz * Axz
DKzxy = - Gmxxz * Axy - Gmyxz * Ayy - Gmzxz * Ayz &
- Gmxyz * Axx - Gmyyz * Axy - Gmzyz * Axz
DKzxz = - Gmxxz * Axz - Gmyxz * Ayz - Gmzxz * Azz &
- Gmxzz * Axx - Gmyzz * Axy - Gmzzz * Axz
DKzyy = - Gmxyz * Axy - Gmyyz * Ayy - Gmzyz * Ayz &
- Gmxyz * Axy - Gmyyz * Ayy - Gmzyz * Ayz
DKzyz = - Gmxyz * Axz - Gmyyz * Ayz - Gmzyz * Azz &
- Gmxzz * Axy - Gmyzz * Ayy - Gmzzz * Ayz
DKzzz = - Gmxzz * Axz - Gmyzz * Ayz - Gmzzz * Azz &
- Gmxzz * Axz - Gmyzz * Ayz - Gmzzz * Azz
!~~~~~~> DKijk = e^(4 phi) (A_jk,i + tmp + 1/3 g_jk K_i)
! - 2 K_ik phi,j + 2 g_ij phiupK_k
! - 2 K_ij phi,k + 2 g_ik phiupK_j
DKxxx = e4phi * (Axxx + DKxxx + F1o3 * gxx * Kx) &
- TWO * Kxx * phix + TWO * gxx * phiupKx &
- TWO * Kxx * phix + TWO * gxx * phiupKx
DKxxy = e4phi * (Axyx + DKxxy + F1o3 * gxy * Kx) &
- TWO * Kxy * phix + TWO * gxx * phiupKy &
- TWO * Kxx * phiy + TWO * gxy * phiupKx
DKxxz = e4phi * (Axzx + DKxxz + F1o3 * gxz * Kx) &
- TWO * Kxz * phix + TWO * gxx * phiupKz &
- TWO * Kxx * phiz + TWO * gxz * phiupKx
DKxyy = e4phi * (Ayyx + DKxyy + F1o3 * gyy * Kx) &
- TWO * Kxy * phiy + TWO * gxy * phiupKy &
- TWO * Kxy * phiy + TWO * gxy * phiupKy
DKxyz = e4phi * (Ayzx + DKxyz + F1o3 * gyz * Kx) &
- TWO * Kxz * phiy + TWO * gxy * phiupKz &
- TWO * Kxy * phiz + TWO * gxz * phiupKy
DKxzz = e4phi * (Azzx + DKxzz + F1o3 * gzz * Kx) &
- TWO * Kxz * phiz + TWO * gxz * phiupKz &
- TWO * Kxz * phiz + TWO * gxz * phiupKz
!~~~~~~>
DKyxx = e4phi * (Axxy + DKyxx + F1o3 * gxx * Ky) &
- TWO * Kxy * phix + TWO * gxy * phiupKx &
- TWO * Kxy * phix + TWO * gxy * phiupKx
DKyxy = e4phi * (Axyy + DKyxy + F1o3 * gxy * Ky) &
- TWO * Kyy * phix + TWO * gxy * phiupKy &
- TWO * Kxy * phiy + TWO * gyy * phiupKx
DKyxz = e4phi * (Axzy + DKyxz + F1o3 * gxz * Ky) &
- TWO * Kyz * phix + TWO * gxy * phiupKz &
- TWO * Kxy * phiz + TWO * gyz * phiupKx
DKyyy = e4phi * (Ayyy + DKyyy + F1o3 * gyy * Ky) &
- TWO * Kyy * phiy + TWO * gyy * phiupKy &
- TWO * Kyy * phiy + TWO * gyy * phiupKy
DKyyz = e4phi * (Ayzy + DKyyz + F1o3 * gyz * Ky) &
- TWO * Kyz * phiy + TWO * gyy * phiupKz &
- TWO * Kyy * phiz + TWO * gyz * phiupKy
DKyzz = e4phi * (Azzy + DKyzz + F1o3 * gzz * Ky) &
- TWO * Kyz * phiz + TWO * gyz * phiupKz &
- TWO * Kyz * phiz + TWO * gyz * phiupKz
!~~~~~~>
DKzxx = e4phi * (Axxz + DKzxx + F1o3 * gxx * Kz) &
- TWO * Kxz * phix + TWO * gxz * phiupKx &
- TWO * Kxz * phix + TWO * gxz * phiupKx
DKzxy = e4phi * (Axyz + DKzxy + F1o3 * gxy * Kz) &
- TWO * Kyz * phix + TWO * gxz * phiupKy &
- TWO * Kxz * phiy + TWO * gyz * phiupKx
DKzxz = e4phi * (Axzz + DKzxz + F1o3 * gxz * Kz) &
- TWO * Kzz * phix + TWO * gxz * phiupKz &
- TWO * Kxz * phiz + TWO * gzz * phiupKx
DKzyy = e4phi * (Ayyz + DKzyy + F1o3 * gyy * Kz) &
- TWO * Kyz * phiy + TWO * gyz * phiupKy &
- TWO * Kyz * phiy + TWO * gyz * phiupKy
DKzyz = e4phi * (Ayzz + DKzyz + F1o3 * gyz * Kz) &
- TWO * Kzz * phiy + TWO * gyz * phiupKz &
- TWO * Kyz * phiz + TWO * gzz * phiupKy
DKzzz = e4phi * (Azzz + DKzzz + F1o3 * gzz * Kz) &
- TWO * Kzz * phiz + TWO * gzz * phiupKz &
- TWO * Kzz * phiz + TWO * gzz * phiupKz
return
end subroutine get_diff_physical_k
!----------------------------------------------------------------------
!------>Begin to compute Psi4
!------>based on quantites:
!------>triad v^i, u^i, w^i
!------>lapse and shift vector beta^i
!------>extrinsic curvature K_ij and trK
!------>covariant derivative of extrinsic curvature D_i K_jk
!------>Ricci tensor: R_ij
!------>gup^ij
!------>Riemann tensor R_ijkl
!----------------------------------------------------------------------
subroutine bssn_compute_psi4(ex, em4phi,lapse, betax,betay,betaz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz, &
vx,vy,vz,ux,uy,uz,wx,wy,wz, &
trK,Kxx,Kxy,Kxz,Kyy,Kyz,Kzz, &
Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz, &
Rxx, Rxy, Rxz, Ryy, Ryz, Rzz, &
DKxxx,DKxxy,DKxxz,DKxyy,DKxyz,DKxzz, &
DKyxx,DKyxy,DKyxz,DKyyy,DKyyz,DKyzz, &
DKzxx,DKzxy,DKzxz,DKzyy,DKzyz,DKzzz, Rpsi4, Ipsi4)
implicit none
!~~~~~~ argument variables
integer,intent(in ):: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: em4phi
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: lapse
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: betax, betay, betaz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gupxx,gupxy,gupxz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gupyy,gupyz,gupzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: vx,vy,vz,ux,uy,uz,wx,wy,wz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: trK,Kxx,Kxy,Kxz,Kyy,Kyz,Kzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Rxyxy, Rxyxz, Rxyyz, Rxzxz, Rxzyz, Ryzyz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: Rxx, Rxy, Rxz, Ryy, Ryz, Rzz
!D_i K_jk ---> DKijk
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: DKxxx,DKxxy,DKxxz,DKxyy,DKxyz,DKxzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: DKyxx,DKyxy,DKyxz,DKyyy,DKyyz,DKyzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: DKzxx,DKzxy,DKzxz,DKzyy,DKzyz,DKzzz
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: Rpsi4,Ipsi4
!~~~~~~ local variables
!n^i upto 1/sqrt(2)
real*8, dimension(ex(1),ex(2),ex(3)) :: nx,ny,nz
!n^i * n^k upto 1/2
real*8, dimension(ex(1),ex(2),ex(3)) :: nnxx,nnxy,nnxz,nnyy,nnyz,nnzz
!u^j * u^l - w^j * w^l
real*8, dimension(ex(1),ex(2),ex(3)) :: uuwwxx,uuwwxy,uuwwxz,uuwwyy,uuwwyz,uuwwzz
!- u^j * w^l - w^j * u^l
real*8, dimension(ex(1),ex(2),ex(3)) :: uwxx,uwxy,uwxz,uwyy,uwyz,uwzz
! temp variables
real*8, dimension(ex(1),ex(2),ex(3)) ::temRxx, temRxy, temRxz, temRyy, temRyz, temRzz
real*8, dimension(ex(1),ex(2),ex(3)) ::temRxyxy,temRxyxz,temRxyyz,temRxzxz,temRxzyz,temRyzyz
real*8, dimension(ex(1),ex(2),ex(3)) ::lapse2
! K^i_j
real*8, dimension(ex(1),ex(2),ex(3)) ::Kupxx,Kupxy,Kupxz,Kupyy,Kupyz,Kupzz
real*8, parameter :: TWO = 2.d0, F1o4 = 1.d0/4.d0
!~~~~~~
! compute n^i = - beta^i/lapse - v^i
nx = - betax/lapse - vx
ny = - betay/lapse - vy
nz = - betaz/lapse - vz
! compute nn^ij = n^i * n^j
nnxx = nx * nx
nnxy = nx * ny
nnxz = nx * nz
nnyy = ny * ny
nnyz = ny * nz
nnzz = nz * nz
! compute uuww^ij = u^i * u^j - w^i * w^j
uuwwxx = ux * ux - wx * wx
uuwwxy = ux * uy - wx * wy
uuwwxz = ux * uz - wx * wz
uuwwyy = uy * uy - wy * wy
uuwwyz = uy * uz - wy * wz
uuwwzz = uz * uz - wz * wz
! compute uw^ij = - u^i * w^j - w^i * u^j
uwxx = ux * wx + wx * ux
uwxy = ux * wy + wx * uy
uwxz = ux * wz + wx * uz
uwyy = uy * wy + wy * uy
uwyz = uy * wz + wy * uz
uwzz = uz * wz + wz * uz
!Commonterm_jl = -1/4 * ( (R_ijkl + K_ik * K_jl - K_il * K_jk) * nn^ik
! - 2 * (D_l K_jk - D_k K_jl) * n^0 * n^k
! + (R_jl - K_jm * K^m_l + K * K_jl) * n^0 * n^0
! )
!add trK * K_jl to R_jl
temRxx = Rxx + trK * Kxx
temRxy = Rxy + trK * Kxy
temRxz = Rxz + trK * Kxz
temRyy = Ryy + trK * Kyy
temRyz = Ryz + trK * Kyz
temRzz = Rzz + trK * Kzz
!add - K_jm * K^m_l to R_jl
! compute K^m_l
call raise(ex,Kxx,Kxy,Kxz,Kupxx,Kupxy,Kupxz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz)
call raise(ex,Kxy,Kyy,Kyz,Kupxy,Kupyy,Kupyz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz)
call raise(ex,Kxz,Kyz,Kzz,Kupxz,Kupyz,Kupzz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz)
temRxx = temRxx - em4phi * ( Kupxx * Kxx + Kupxy * Kxy + Kupxz * Kxz )
temRxy = temRxy - em4phi * ( Kupxx * Kxy + Kupxy * Kyy + Kupxz * Kyz )
temRxz = temRxz - em4phi * ( Kupxx * Kxz + Kupxy * Kyz + Kupxz * Kzz )
temRyy = temRyy - em4phi * ( Kupxy * Kxy + Kupyy * Kyy + Kupyz * Kyz )
temRyz = temRyz - em4phi * ( Kupxy * Kxz + Kupyy * Kyz + Kupyz * Kzz )
temRzz = temRzz - em4phi * ( Kupxz * Kxz + Kupyz * Kyz + Kupzz * Kzz )
! multiply with n^0 * n^0 upto 1/2
! n^0 = 1/(sqrt(2) * lapse)
lapse2 = lapse * lapse
temRxx = temRxx/lapse2
temRxy = temRxy/lapse2
temRxz = temRxz/lapse2
temRyy = temRyy/lapse2
temRyz = temRyz/lapse2
temRzz = temRzz/lapse2
!add (K_ik * K_jl - K_il * K_jk) to R_ijkl, note they have the same symmetric index
temRxyxy = Rxyxy + Kxx * Kyy - Kxy * Kxy
temRxyxz = Rxyxz + Kxx * Kyz - Kxz * Kxy
temRxyyz = Rxyyz + Kxy * Kyz - Kxz * Kyy
temRxzxz = Rxzxz + Kxx * Kzz - Kxz * Kxz
temRxzyz = Rxzyz + Kxy * Kzz - Kxz * Kyz
temRyzyz = Ryzyz + Kyy * Kzz - Kyz * Kyz
!add (R_ijkl + K_ik * K_jl - K_il * K_jk) * nn^ik to R_jl, upto 1/2
! note they have the same symmetric index
temRxx = temRxx + temRxyxy * nnyy + temRxyxz * nnyz + temRxyxz * nnyz + temRxzxz * nnzz
temRxy = temRxy - temRxyxy * nnxy + temRxyyz * nnyz - temRxyxz * nnxz + temRxzyz * nnzz
temRxz = temRxz - temRxyxz * nnxy - temRxyyz * nnyy - temRxzxz * nnxz - temRxzyz * nnyz
temRyy = temRyy + temRxyxy * nnxx - temRxyyz * nnxz - temRxyyz * nnxz + temRyzyz * nnzz
temRyz = temRyz + temRxyxz * nnxx + temRxyyz * nnxy - temRxzyz * nnxz - temRyzyz * nnyz
temRzz = temRzz + temRxzxz * nnxx + temRxzyz * nnxy + temRxzyz * nnxy + temRyzyz * nnyy
!add 2 * (D_k K_jl * n^0 * n^k) to R_jl, upto 1/2
temRxx = temRxx + TWO * ( DKxxx * nx + DKyxx * ny + DKzxx * nz)/lapse
temRxy = temRxy + TWO * ( DKxxy * nx + DKyxy * ny + DKzxy * nz)/lapse
temRxz = temRxz + TWO * ( DKxxz * nx + DKyxz * ny + DKzxz * nz)/lapse
temRyy = temRyy + TWO * ( DKxyy * nx + DKyyy * ny + DKzyy * nz)/lapse
temRyz = temRyz + TWO * ( DKxyz * nx + DKyyz * ny + DKzyz * nz)/lapse
temRzz = temRzz + TWO * ( DKxzz * nx + DKyzz * ny + DKzzz * nz)/lapse
!add - (D_l K_jk + D_j K_lk) * n^0 * ^k to R_jl, upto 1/2
! note we symmetrize the index here
temRxx = temRxx - ((DKxxx + DKxxx) * nx + (DKxxy + DKxxy) * ny + (DKxxz + DKxxz) * nz)/lapse
temRxy = temRxy - ((DKyxx + DKxxy) * nx + (DKyxy + DKxyy) * ny + (DKyxz + DKxyz) * nz)/lapse
temRxz = temRxz - ((DKzxx + DKxxz) * nx + (DKzxy + DKxyz) * ny + (DKzxz + DKxzz) * nz)/lapse
temRyy = temRyy - ((DKyxy + DKyxy) * nx + (DKyyy + DKyyy) * ny + (DKyyz + DKyyz) * nz)/lapse
temRyz = temRyz - ((DKzxy + DKyxz) * nx + (DKzyy + DKyyz) * ny + (DKzyz + DKyzz) * nz)/lapse
temRzz = temRzz - ((DKzxz + DKzxz) * nx + (DKzyz + DKzyz) * ny + (DKzzz + DKzzz) * nz)/lapse
!the real part of Psi4
Rpsi4 = temRxx * uuwwxx + temRyy * uuwwyy + temRzz * uuwwzz &
+ (temRxy * uuwwxy + temRxz * uuwwxz + temRyz * uuwwyz) * TWO
!the imaginary part of Psi4
Ipsi4 = temRxx * uwxx + temRyy * uwyy + temRzz * uwzz &
+ (temRxy * uwxy + temRxz * uwxz + temRyz * uwyz) * TWO
!multiply with -1/4
Rpsi4 = - F1o4 * Rpsi4
Ipsi4 = - F1o4 * Ipsi4
return
end subroutine bssn_compute_psi4
!-----------------------------------------------------------------------------
! covariant derivatives w.r.t *tilded metric* of *symmetric* variable
!-----------------------------------------------------------------------------
subroutine xcov_deriv(ex,fx,fy,fz,fxx,fxy,fxz,fyy,fyz,fzz, &
Gamxxx, Gamxxy, Gamxxz, Gamxyy, Gamxyz, Gamxzz, &
Gamyxx, Gamyxy, Gamyxz, Gamyyy, Gamyyz, Gamyzz, &
Gamzxx, Gamzxy, Gamzxz, Gamzyy, Gamzyz, Gamzzz)
implicit none
!~~~~~~ Input arguments
integer, intent(in ) :: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: fx, fy, fz
real*8, dimension(ex(1),ex(2),ex(3)), intent(out) :: fxx,fxy,fxz,fyy,fyz,fzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gamxxx, Gamxxy, Gamxxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gamxyy, Gamxyz, Gamxzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gamyxx, Gamyxy, Gamyxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gamyyy, Gamyyz, Gamyzz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gamzxx, Gamzxy, Gamzxz
real*8, dimension(ex(1),ex(2),ex(3)), intent(in ) :: Gamzyy, Gamzyz, Gamzzz
!~~~~~~ Add Connection terms
fxx = fxx - Gamxxx * fx - Gamyxx * fy - Gamzxx * fz
fxy = fxy - Gamxxy * fx - Gamyxy * fy - Gamzxy * fz
fxz = fxz - Gamxxz * fx - Gamyxz * fy - Gamzxz * fz
fyy = fyy - Gamxyy * fx - Gamyyy * fy - Gamzyy * fz
fyz = fyz - Gamxyz * fx - Gamyyz * fy - Gamzyz * fz
fzz = fzz - Gamxzz * fx - Gamyzz * fy - Gamzzz * fz
return
end subroutine xcov_deriv
!--------------------------------------------------------------------
! Gram-Schmidt orthonormal in Cartesin coordinate
! V1 = [ x, y, z ]
! V2 = [-y, x, ZEO]
! V3 = [xz,yz,-(x^2+y^2)]
! V1 -> V1 / sqrt(W11)
! V2 -> ( V2 - V1 * W12 ) / sqrt(W22)
! V3 -> ( V3 - V1 * W13 - V2 * W23 ) / sqrt(W33)
! W_ij = g_ab * Vi^a * Vj^b
! it is metric, not tilde metric
! V1 first
!--------------------------------------------------------------------
subroutine get_triad0(ex, X, Y, Z, ep4phi, &
gxxi,gxyi,gxzi,gyyi,gyzi,gzzi, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
implicit none
!~~~~~~ argument variables
integer,intent(in ):: ex(1:3)
real*8, intent(in ):: X(1:ex(1)),Y(1:ex(2)),Z(1:ex(3))
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ep4phi !exp(4 * phi)
! tilted metric
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gxxi,gxyi,gxzi,gyyi,gyzi,gzzi
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: vx,vy,vz,ux,uy,uz,wx,wy,wz
!~~~~~~ local variables
real*8, dimension(ex(1),ex(2),ex(3)) :: norm
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gxy,gxz,gyy,gyz,gzz
real*8,parameter:: ZEO = 0.d0, ONE = 1.d0
integer::i,j,k
!~~~~~~
gxx = gxxi * ep4phi
gxy = gxyi * ep4phi
gxz = gxzi * ep4phi
gyy = gyyi * ep4phi
gyz = gyzi * ep4phi
gzz = gzzi * ep4phi
! initialize U, V, W vetors
do i=1,ex(1)
do j=1,ex(2)
do k=1,ex(3)
vx(i,j,k) = X(i)
vy(i,j,k) = Y(j)
vz(i,j,k) = Z(k)
ux(i,j,k) = - Y(j)
uy(i,j,k) = X(i)
uz(i,j,k) = ZEO
wx(i,j,k) = X(i)*Z(k)
wy(i,j,k) = Y(j)*Z(k)
wz(i,j,k) = -(X(i)*X(i) + Y(j)*Y(j))
enddo
enddo
enddo
! Gram-Schmidt orthonormalization
call InnerProd(ex,norm,vx,vy,vz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
vx = vx/dsqrt(norm)
vy = vy/dsqrt(norm)
vz = vz/dsqrt(norm)
call InnerProd(ex,norm,ux,uy,uz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux - norm*vx
uy = uy - norm*vy
uz = uz - norm*vz
call InnerProd(ex,norm,ux,uy,uz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux/dsqrt(norm)
uy = uy/dsqrt(norm)
uz = uz/dsqrt(norm)
call InnerProd(ex,norm,wx,wy,wz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*vx
wy = wy - norm*vy
wz = wz - norm*vz
call InnerProd(ex,norm,wx,wy,wz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*ux
wy = wy - norm*uy
wz = wz - norm*uz
call InnerProd(ex,norm,wx,wy,wz,wx,wy,wz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx/dsqrt(norm)
wy = wy/dsqrt(norm)
wz = wz/dsqrt(norm)
return
end subroutine get_triad0
!--------------------------------------------------------------------
! Gram-Schmidt orthonormal in Cartesin coordinate
! V1 = [ x, y, z ]
! V2 = [-y, x, ZEO]
! V3 = [xz,yz,-(x^2+y^2)]
! V1 -> V1 / sqrt(W11)
! V2 -> ( V2 - V1 * W12 ) / sqrt(W22)
! V3 -> ( V3 - V1 * W13 - V2 * W23 ) / sqrt(W33)
! W_ij = g_ab * Vi^a * Vj^b
! it is metric, not tilde metric
! V2 first
!--------------------------------------------------------------------
subroutine get_triad1(ex, X, Y, Z, ep4phi, &
gxxi,gxyi,gxzi,gyyi,gyzi,gzzi, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
implicit none
!~~~~~~ argument variables
integer,intent(in ):: ex(1:3)
real*8, intent(in ):: X(1:ex(1)),Y(1:ex(2)),Z(1:ex(3))
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ep4phi !exp(4 * phi)
! tilted metric
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gxxi,gxyi,gxzi,gyyi,gyzi,gzzi
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: vx,vy,vz,ux,uy,uz,wx,wy,wz
!~~~~~~ local variables
real*8, dimension(ex(1),ex(2),ex(3)) :: norm
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gxy,gxz,gyy,gyz,gzz
real*8,parameter:: ZEO = 0.d0, ONE = 1.d0
integer::i,j,k
!~~~~~~
gxx = gxxi * ep4phi
gxy = gxyi * ep4phi
gxz = gxzi * ep4phi
gyy = gyyi * ep4phi
gyz = gyzi * ep4phi
gzz = gzzi * ep4phi
! initialize U, V, W vetors
do i=1,ex(1)
do j=1,ex(2)
do k=1,ex(3)
vx(i,j,k) = X(i)
vy(i,j,k) = Y(j)
vz(i,j,k) = Z(k)
ux(i,j,k) = - Y(j)
uy(i,j,k) = X(i)
uz(i,j,k) = ZEO
wx(i,j,k) = X(i)*Z(k)
wy(i,j,k) = Y(j)*Z(k)
wz(i,j,k) = -(X(i)*X(i) + Y(j)*Y(j))
enddo
enddo
enddo
! Gram-Schmidt orthonormalization
call InnerProd(ex,norm,ux,uy,uz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux/dsqrt(norm)
uy = uy/dsqrt(norm)
uz = uz/dsqrt(norm)
call InnerProd(ex,norm,vx,vy,vz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
vx = vx - norm*ux
vy = vy - norm*uy
vz = vz - norm*uz
call InnerProd(ex,norm,vx,vy,vz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
vx = vx/dsqrt(norm)
vy = vy/dsqrt(norm)
vz = vz/dsqrt(norm)
call InnerProd(ex,norm,wx,wy,wz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*ux
wy = wy - norm*uy
wz = wz - norm*uz
call InnerProd(ex,norm,wx,wy,wz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*vx
wy = wy - norm*vy
wz = wz - norm*vz
call InnerProd(ex,norm,wx,wy,wz,wx,wy,wz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx/dsqrt(norm)
wy = wy/dsqrt(norm)
wz = wz/dsqrt(norm)
return
end subroutine get_triad1
!--------------------------------------------------------------------
! Gram-Schmidt orthonormal in Cartesin coordinate
! V1 = [ x, y, z ]
! V2 = [-y, x, ZEO]
! V3 = [xz,yz,-(x^2+y^2)]
! V1 -> V1 / sqrt(W11)
! V2 -> ( V2 - V1 * W12 ) / sqrt(W22)
! V3 -> ( V3 - V1 * W13 - V2 * W23 ) / sqrt(W33)
! W_ij = g_ab * Vi^a * Vj^b
! it is metric, not tilde metric
! raise V1, then V1 first
!--------------------------------------------------------------------
subroutine get_triad2(ex, X, Y, Z, ep4phi, &
gxxi,gxyi,gxzi,gyyi,gyzi,gzzi, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
implicit none
!~~~~~~ argument variables
integer,intent(in ):: ex(1:3)
real*8, intent(in ):: X(1:ex(1)),Y(1:ex(2)),Z(1:ex(3))
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ep4phi !exp(4 * phi)
! tilted metric
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gxxi,gxyi,gxzi,gyyi,gyzi,gzzi
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: vx,vy,vz,ux,uy,uz,wx,wy,wz
!~~~~~~ local variables
real*8, dimension(ex(1),ex(2),ex(3)) :: norm,fx,fy,fz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gxy,gxz,gyy,gyz,gzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gupxx,gupxy,gupxz,gupyy,gupyz,gupzz
real*8,parameter:: ZEO = 0.d0, ONE = 1.d0
integer::i,j,k
!~~~~~~
gxx = gxxi * ep4phi
gxy = gxyi * ep4phi
gxz = gxzi * ep4phi
gyy = gyyi * ep4phi
gyz = gyzi * ep4phi
gzz = gzzi * ep4phi
! invert metric
gupzz = gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
gupxx = ( gyy * gzz - gyz * gyz ) / gupzz
gupxy = - ( gxy * gzz - gyz * gxz ) / gupzz
gupxz = ( gxy * gyz - gyy * gxz ) / gupzz
gupyy = ( gxx * gzz - gxz * gxz ) / gupzz
gupyz = - ( gxx * gyz - gxy * gxz ) / gupzz
gupzz = ( gxx * gyy - gxy * gxy ) / gupzz
! initialize U, V, W vetors
do i=1,ex(1)
do j=1,ex(2)
do k=1,ex(3)
vx(i,j,k) = X(i)
vy(i,j,k) = Y(j)
vz(i,j,k) = Z(k)
ux(i,j,k) = - Y(j)
uy(i,j,k) = X(i)
uz(i,j,k) = ZEO
wx(i,j,k) = X(i)*Z(k)
wy(i,j,k) = Y(j)*Z(k)
wz(i,j,k) = -(X(i)*X(i) + Y(j)*Y(j))
enddo
enddo
enddo
fx = vx
fy = vy
fz = vz
call raise(ex,fx,fy,fz,vx,vy,vz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz)
! Gram-Schmidt orthonormalization
call InnerProd(ex,norm,vx,vy,vz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
vx = vx/dsqrt(norm)
vy = vy/dsqrt(norm)
vz = vz/dsqrt(norm)
call InnerProd(ex,norm,ux,uy,uz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux - norm*vx
uy = uy - norm*vy
uz = uz - norm*vz
call InnerProd(ex,norm,ux,uy,uz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux/dsqrt(norm)
uy = uy/dsqrt(norm)
uz = uz/dsqrt(norm)
call InnerProd(ex,norm,wx,wy,wz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*vx
wy = wy - norm*vy
wz = wz - norm*vz
call InnerProd(ex,norm,wx,wy,wz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*ux
wy = wy - norm*uy
wz = wz - norm*uz
call InnerProd(ex,norm,wx,wy,wz,wx,wy,wz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx/dsqrt(norm)
wy = wy/dsqrt(norm)
wz = wz/dsqrt(norm)
return
end subroutine get_triad2
!***********for shell*********************
!--------------------------------------------------------------------
! Gram-Schmidt orthonormal in Cartesin coordinate
! V1 = [ x, y, z ]
! V2 = [-y, x, ZEO]
! V3 = [xz,yz,-(x^2+y^2)]
! V1 -> V1 / sqrt(W11)
! V2 -> ( V2 - V1 * W12 ) / sqrt(W22)
! V3 -> ( V3 - V1 * W13 - V2 * W23 ) / sqrt(W33)
! W_ij = g_ab * Vi^a * Vj^b
! it is metric, not tilde metric
! V1 first
!--------------------------------------------------------------------
subroutine get_triad0_ss(ex, X, Y, Z, ep4phi, &
gxxi,gxyi,gxzi,gyyi,gyzi,gzzi, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
implicit none
!~~~~~~ argument variables
integer,intent(in ):: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: X,Y,Z
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ep4phi !exp(4 * phi)
! tilted metric
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gxxi,gxyi,gxzi,gyyi,gyzi,gzzi
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: vx,vy,vz,ux,uy,uz,wx,wy,wz
!~~~~~~ local variables
real*8, dimension(ex(1),ex(2),ex(3)) :: norm
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gxy,gxz,gyy,gyz,gzz
real*8,parameter:: ZEO = 0.d0, ONE = 1.d0
integer::i,j,k
real*8,parameter::TINYRR=1.d-14
!~~~~~~
gxx = gxxi * ep4phi
gxy = gxyi * ep4phi
gxz = gxzi * ep4phi
gyy = gyyi * ep4phi
gyz = gyzi * ep4phi
gzz = gzzi * ep4phi
! initialize U, V, W vetors
do i=1,ex(1)
do j=1,ex(2)
do k=1,ex(3)
if(abs(X(i,j,k)) < TINYRR .and. abs(Y(i,j,k)) < TINYRR .and. abs(Z(i,j,k)) < TINYRR)then
vx(i,j,k) = TINYRR
vy(i,j,k) = TINYRR
vz(i,j,k) = TINYRR
else
vx(i,j,k) = X(i,j,k)
vy(i,j,k) = Y(i,j,k)
vz(i,j,k) = Z(i,j,k)
endif
if(abs(X(i,j,k)) < TINYRR .and. abs(Y(i,j,k)) < TINYRR)then
ux(i,j,k) = - TINYRR
uy(i,j,k) = TINYRR
uz(i,j,k) = ZEO
wx(i,j,k) = TINYRR*Z(i,j,k)
wy(i,j,k) = TINYRR*Z(i,j,k)
wz(i,j,k) = -2*TINYRR*TINYRR
else
ux(i,j,k) = - Y(i,j,k)
uy(i,j,k) = X(i,j,k)
uz(i,j,k) = ZEO
wx(i,j,k) = X(i,j,k)*Z(i,j,k)
wy(i,j,k) = Y(i,j,k)*Z(i,j,k)
wz(i,j,k) = -(X(i,j,k)*X(i,j,k) + Y(i,j,k)*Y(i,j,k))
endif
enddo
enddo
enddo
! Gram-Schmidt orthonormalization
call InnerProd(ex,norm,vx,vy,vz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
vx = vx/dsqrt(norm)
vy = vy/dsqrt(norm)
vz = vz/dsqrt(norm)
call InnerProd(ex,norm,ux,uy,uz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux - norm*vx
uy = uy - norm*vy
uz = uz - norm*vz
call InnerProd(ex,norm,ux,uy,uz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux/dsqrt(norm)
uy = uy/dsqrt(norm)
uz = uz/dsqrt(norm)
call InnerProd(ex,norm,wx,wy,wz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*vx
wy = wy - norm*vy
wz = wz - norm*vz
call InnerProd(ex,norm,wx,wy,wz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*ux
wy = wy - norm*uy
wz = wz - norm*uz
call InnerProd(ex,norm,wx,wy,wz,wx,wy,wz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx/dsqrt(norm)
wy = wy/dsqrt(norm)
wz = wz/dsqrt(norm)
return
end subroutine get_triad0_ss
!--------------------------------------------------------------------
! Gram-Schmidt orthonormal in Cartesin coordinate
! V1 = [ x, y, z ]
! V2 = [-y, x, ZEO]
! V3 = [xz,yz,-(x^2+y^2)]
! V1 -> V1 / sqrt(W11)
! V2 -> ( V2 - V1 * W12 ) / sqrt(W22)
! V3 -> ( V3 - V1 * W13 - V2 * W23 ) / sqrt(W33)
! W_ij = g_ab * Vi^a * Vj^b
! it is metric, not tilde metric
! V2 first
!--------------------------------------------------------------------
subroutine get_triad1_ss(ex, X, Y, Z, ep4phi, &
gxxi,gxyi,gxzi,gyyi,gyzi,gzzi, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
implicit none
!~~~~~~ argument variables
integer,intent(in ):: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: X,Y,Z
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ep4phi !exp(4 * phi)
! tilted metric
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gxxi,gxyi,gxzi,gyyi,gyzi,gzzi
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: vx,vy,vz,ux,uy,uz,wx,wy,wz
!~~~~~~ local variables
real*8, dimension(ex(1),ex(2),ex(3)) :: norm
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gxy,gxz,gyy,gyz,gzz
real*8,parameter:: ZEO = 0.d0, ONE = 1.d0
integer::i,j,k
real*8,parameter::TINYRR=1.d-14
!~~~~~~
gxx = gxxi * ep4phi
gxy = gxyi * ep4phi
gxz = gxzi * ep4phi
gyy = gyyi * ep4phi
gyz = gyzi * ep4phi
gzz = gzzi * ep4phi
! initialize U, V, W vetors
do i=1,ex(1)
do j=1,ex(2)
do k=1,ex(3)
if(abs(X(i,j,k)) < TINYRR .and. abs(Y(i,j,k)) < TINYRR .and. abs(Z(i,j,k)) < TINYRR)then
vx(i,j,k) = TINYRR
vy(i,j,k) = TINYRR
vz(i,j,k) = TINYRR
else
vx(i,j,k) = X(i,j,k)
vy(i,j,k) = Y(i,j,k)
vz(i,j,k) = Z(i,j,k)
endif
if(abs(X(i,j,k)) < TINYRR .and. abs(Y(i,j,k)) < TINYRR)then
ux(i,j,k) = - TINYRR
uy(i,j,k) = TINYRR
uz(i,j,k) = ZEO
wx(i,j,k) = TINYRR*Z(i,j,k)
wy(i,j,k) = TINYRR*Z(i,j,k)
wz(i,j,k) = -2*TINYRR*TINYRR
else
ux(i,j,k) = - Y(i,j,k)
uy(i,j,k) = X(i,j,k)
uz(i,j,k) = ZEO
wx(i,j,k) = X(i,j,k)*Z(i,j,k)
wy(i,j,k) = Y(i,j,k)*Z(i,j,k)
wz(i,j,k) = -(X(i,j,k)*X(i,j,k) + Y(i,j,k)*Y(i,j,k))
endif
enddo
enddo
enddo
! Gram-Schmidt orthonormalization
call InnerProd(ex,norm,ux,uy,uz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux/dsqrt(norm)
uy = uy/dsqrt(norm)
uz = uz/dsqrt(norm)
call InnerProd(ex,norm,vx,vy,vz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
vx = vx - norm*ux
vy = vy - norm*uy
vz = vz - norm*uz
call InnerProd(ex,norm,vx,vy,vz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
vx = vx/dsqrt(norm)
vy = vy/dsqrt(norm)
vz = vz/dsqrt(norm)
call InnerProd(ex,norm,wx,wy,wz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*ux
wy = wy - norm*uy
wz = wz - norm*uz
call InnerProd(ex,norm,wx,wy,wz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*vx
wy = wy - norm*vy
wz = wz - norm*vz
call InnerProd(ex,norm,wx,wy,wz,wx,wy,wz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx/dsqrt(norm)
wy = wy/dsqrt(norm)
wz = wz/dsqrt(norm)
return
end subroutine get_triad1_ss
!--------------------------------------------------------------------
! Gram-Schmidt orthonormal in Cartesin coordinate
! V1 = [ x, y, z ]
! V2 = [-y, x, ZEO]
! V3 = [xz,yz,-(x^2+y^2)]
! V1 -> V1 / sqrt(W11)
! V2 -> ( V2 - V1 * W12 ) / sqrt(W22)
! V3 -> ( V3 - V1 * W13 - V2 * W23 ) / sqrt(W33)
! W_ij = g_ab * Vi^a * Vj^b
! it is metric, not tilde metric
! raise V1, then V1 first
!--------------------------------------------------------------------
subroutine get_triad2_ss(ex, X, Y, Z, ep4phi, &
gxxi,gxyi,gxzi,gyyi,gyzi,gzzi, &
vx,vy,vz,ux,uy,uz,wx,wy,wz)
implicit none
!~~~~~~ argument variables
integer,intent(in ):: ex(1:3)
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: X,Y,Z
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: ep4phi !exp(4 * phi)
! tilted metric
real*8, dimension(ex(1),ex(2),ex(3)),intent(in ) :: gxxi,gxyi,gxzi,gyyi,gyzi,gzzi
real*8, dimension(ex(1),ex(2),ex(3)),intent(out) :: vx,vy,vz,ux,uy,uz,wx,wy,wz
!~~~~~~ local variables
real*8, dimension(ex(1),ex(2),ex(3)) :: norm,fx,fy,fz
real*8, dimension(ex(1),ex(2),ex(3)) :: gxx,gxy,gxz,gyy,gyz,gzz
real*8, dimension(ex(1),ex(2),ex(3)) :: gupxx,gupxy,gupxz,gupyy,gupyz,gupzz
real*8,parameter:: ZEO = 0.d0, ONE = 1.d0
integer::i,j,k
real*8,parameter::TINYRR=1.d-14
!~~~~~~
gxx = gxxi * ep4phi
gxy = gxyi * ep4phi
gxz = gxzi * ep4phi
gyy = gyyi * ep4phi
gyz = gyzi * ep4phi
gzz = gzzi * ep4phi
! invert metric
gupzz = gxx * gyy * gzz + gxy * gyz * gxz + gxz * gxy * gyz - &
gxz * gyy * gxz - gxy * gxy * gzz - gxx * gyz * gyz
gupxx = ( gyy * gzz - gyz * gyz ) / gupzz
gupxy = - ( gxy * gzz - gyz * gxz ) / gupzz
gupxz = ( gxy * gyz - gyy * gxz ) / gupzz
gupyy = ( gxx * gzz - gxz * gxz ) / gupzz
gupyz = - ( gxx * gyz - gxy * gxz ) / gupzz
gupzz = ( gxx * gyy - gxy * gxy ) / gupzz
! initialize U, V, W vetors
do i=1,ex(1)
do j=1,ex(2)
do k=1,ex(3)
if(abs(X(i,j,k)) < TINYRR .and. abs(Y(i,j,k)) < TINYRR .and. abs(Z(i,j,k)) < TINYRR)then
vx(i,j,k) = TINYRR
vy(i,j,k) = TINYRR
vz(i,j,k) = TINYRR
else
vx(i,j,k) = X(i,j,k)
vy(i,j,k) = Y(i,j,k)
vz(i,j,k) = Z(i,j,k)
endif
if(abs(X(i,j,k)) < TINYRR .and. abs(Y(i,j,k)) < TINYRR)then
ux(i,j,k) = - TINYRR
uy(i,j,k) = TINYRR
uz(i,j,k) = ZEO
wx(i,j,k) = TINYRR*Z(i,j,k)
wy(i,j,k) = TINYRR*Z(i,j,k)
wz(i,j,k) = -2*TINYRR*TINYRR
else
ux(i,j,k) = - Y(i,j,k)
uy(i,j,k) = X(i,j,k)
uz(i,j,k) = ZEO
wx(i,j,k) = X(i,j,k)*Z(i,j,k)
wy(i,j,k) = Y(i,j,k)*Z(i,j,k)
wz(i,j,k) = -(X(i,j,k)*X(i,j,k) + Y(i,j,k)*Y(i,j,k))
endif
enddo
enddo
enddo
fx = vx
fy = vy
fz = vz
call raise(ex,fx,fy,fz,vx,vy,vz, &
gupxx,gupxy,gupxz,gupyy,gupyz,gupzz)
! Gram-Schmidt orthonormalization
call InnerProd(ex,norm,vx,vy,vz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
vx = vx/dsqrt(norm)
vy = vy/dsqrt(norm)
vz = vz/dsqrt(norm)
call InnerProd(ex,norm,ux,uy,uz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux - norm*vx
uy = uy - norm*vy
uz = uz - norm*vz
call InnerProd(ex,norm,ux,uy,uz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
ux = ux/dsqrt(norm)
uy = uy/dsqrt(norm)
uz = uz/dsqrt(norm)
call InnerProd(ex,norm,wx,wy,wz,vx,vy,vz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*vx
wy = wy - norm*vy
wz = wz - norm*vz
call InnerProd(ex,norm,wx,wy,wz,ux,uy,uz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx - norm*ux
wy = wy - norm*uy
wz = wz - norm*uz
call InnerProd(ex,norm,wx,wy,wz,wx,wy,wz,gxx,gxy,gxz,gyy,gyz,gzz)
wx = wx/dsqrt(norm)
wy = wy/dsqrt(norm)
wz = wz/dsqrt(norm)
return
end subroutine get_triad2_ss
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