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C @@
C @file Gowdy.F77
C @date December 2002
C @author Denis Pollney
C @desc
C Cosmological Gowdy metric for a polarized wave in an
C expanding universe. See
C "Stable 3-level leapfrog integration in numerical relativity"
C New, K, Watt, K, Misner, C and Centrella, J, PRD 58, 064022.
C @desc
C @version $Header$
C @@
#include "cctk.h"
#include "cctk_Parameters.h"
subroutine Exact__Gowdy_wave(
$ x, y, z, t,
$ gdtt, gdtx, gdty, gdtz,
$ gdxx, gdyy, gdzz, gdxy, gdyz, gdzx,
$ gutt, gutx, guty, gutz,
$ guxx, guyy, guzz, guxy, guyz, guzx,
$ psi, Tmunu_flag)
implicit none
DECLARE_CCTK_PARAMETERS
c input arguments
CCTK_REAL x, y, z, t
c output arguments
CCTK_REAL gdtt, gdtx, gdty, gdtz,
$ gdxx, gdyy, gdzz, gdxy, gdyz, gdzx,
$ gutt, gutx, guty, gutz,
$ guxx, guyy, guzz, guxy, guyz, guzx
CCTK_REAL psi
LOGICAL Tmunu_flag
c local static variables
logical firstcall
CCTK_REAL amp
CCTK_REAL PI, twoPI
CCTK_REAL Bessel_J0, Bessel_J1
data firstcall /.true./
save firstcall, amp, PI, twoPI, Bessel_J0, Bessel_J1
c$omp threadprivate (firstcall, amp, PI, twoPI, Bessel_J0, Bessel_J1)
c local variables
CCTK_REAL Bessel_J0_t, Bessel_J1_t
CCTK_REAL cosz, eP, lambda
CCTK_REAL d1, d2, d3, d4, d5, d6
C This is a vacuum spacetime with no cosmological constant
Tmunu_flag = .false.
if (firstcall) then
amp = Gowdy_wave__amplitude
PI = acos(-1.d0)
twoPI = 2.d0*PI
call jy01a(twoPI, Bessel_J0, d1, Bessel_J1, d2, d3, d4, d5, d6)
firstcall = .false.
end if
if (t.eq.0.d0) then
call CCTK_WARN(0, "The Gowdy metric is singular for t=0. You may need to set cactus::cctk_initial_time > 0.")
end if
call jy01a(twoPI*t, Bessel_J0_t, d1, Bessel_J1_t, d2, d3, d4, d5, d6)
cosz = cos(twoPI*z)
eP = exp(Bessel_J0_t * cosz)
lambda = amp * (-twoPI * t * Bessel_J0_t * Bessel_J1_t * cosz**2
+ + twoPI * PI * t**2 * (Bessel_J0_t**2 + Bessel_J1_t**2)
+ - 0.5d0 * (twoPI**2 * (Bessel_J0**2 + Bessel_J1**2)
+ - twoPI * Bessel_J0 * Bessel_J1))
c
c lower metric components.
c
gdtt = -exp(0.5d0*lambda)/sqrt(t)
gdtx = 0.d0
gdty = 0.d0
gdtz = 0.d0
gdxx = t * eP
gdyy = t / eP
gdzz = -gdtt
gdxy = 0.d0
gdyz = 0.d0
gdzx = 0.d0
c
c upper metric components.
c
gutt = 1.d0 / gdtt
gutx = 0.d0
guty = 0.d0
gutz = 0.d0
guxx = 1.d0 / gdxx
guyy = 1.d0 / gdyy
guzz = 1.d0 / gdzz
guxy = 0.d0
guyz = 0.d0
guzx = 0.d0
return
end
C @@
C @routine Bessel.F77
C @date December 2002
C @author Denis Pollney
C @desc
C Compute bessel functions of 0th and 1st order.
C This routine was downloaded from:
C http://iris-lee3.ece.uiuc.edu/~jjin/routines/routines.html
C
C =======================================================
C Purpose: Compute Bessel functions J0(x), J1(x), Y0(x),
C Y1(x), and their derivatives
C Input : x --- Argument of Jn(x) & Yn(x) ( x ? 0 )
C Output: BJ0 --- J0(x)
C DJ0 --- dJ0(x)
C BJ1 --- J1(x)
C DJ1 --- dJ1(x)
C BY0 --- Y0(x)
C DY0 --- dY0(x)
C BY1 --- Y1(x)
C DY1 --- dY1(x)
C =======================================================
C
C @desc
C @version $Header$
C @@
SUBROUTINE JY01A(X,BJ0,DJ0,BJ1,DJ1,BY0,DY0,BY1,DY1)
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION A(12),B(12),A1(12),B1(12)
PI=3.141592653589793D0
RP2=0.63661977236758D0
X2=X*X
IF (X.EQ.0.0D0) THEN
BJ0=1.0D0
BJ1=0.0D0
DJ0=0.0D0
DJ1=0.5D0
BY0=-1.0D+300
BY1=-1.0D+300
DY0=1.0D+300
DY1=1.0D+300
RETURN
ENDIF
IF (X.LE.12.0D0) THEN
BJ0=1.0D0
R=1.0D0
DO 5 K=1,30
R=-0.25D0*R*X2/(K*K)
BJ0=BJ0+R
IF (DABS(R).LT.DABS(BJ0)*1.0D-15) GO TO 10
5 CONTINUE
10 BJ1=1.0D0
R=1.0D0
DO 15 K=1,30
R=-0.25D0*R*X2/(K*(K+1.0D0))
BJ1=BJ1+R
IF (DABS(R).LT.DABS(BJ1)*1.0D-15) GO TO 20
15 CONTINUE
20 BJ1=0.5D0*X*BJ1
EC=DLOG(X/2.0D0)+0.5772156649015329D0
CS0=0.0D0
W0=0.0D0
R0=1.0D0
DO 25 K=1,30
W0=W0+1.0D0/K
R0=-0.25D0*R0/(K*K)*X2
R=R0*W0
CS0=CS0+R
IF (DABS(R).LT.DABS(CS0)*1.0D-15) GO TO 30
25 CONTINUE
30 BY0=RP2*(EC*BJ0-CS0)
CS1=1.0D0
W1=0.0D0
R1=1.0D0
DO 35 K=1,30
W1=W1+1.0D0/K
R1=-0.25D0*R1/(K*(K+1))*X2
R=R1*(2.0D0*W1+1.0D0/(K+1.0D0))
CS1=CS1+R
IF (DABS(R).LT.DABS(CS1)*1.0D-15) GO TO 40
35 CONTINUE
40 BY1=RP2*(EC*BJ1-1.0D0/X-0.25D0*X*CS1)
ELSE
DATA A/-.7031250000000000D-01,.1121520996093750D+00,
& -.5725014209747314D+00,.6074042001273483D+01,
& -.1100171402692467D+03,.3038090510922384D+04,
& -.1188384262567832D+06,.6252951493434797D+07,
& -.4259392165047669D+09,.3646840080706556D+11,
& -.3833534661393944D+13,.4854014686852901D+15/
DATA B/ .7324218750000000D-01,-.2271080017089844D+00,
& .1727727502584457D+01,-.2438052969955606D+02,
& .5513358961220206D+03,-.1825775547429318D+05,
& .8328593040162893D+06,-.5006958953198893D+08,
& .3836255180230433D+10,-.3649010818849833D+12,
& .4218971570284096D+14,-.5827244631566907D+16/
DATA A1/.1171875000000000D+00,-.1441955566406250D+00,
& .6765925884246826D+00,-.6883914268109947D+01,
& .1215978918765359D+03,-.3302272294480852D+04,
& .1276412726461746D+06,-.6656367718817688D+07,
& .4502786003050393D+09,-.3833857520742790D+11,
& .4011838599133198D+13,-.5060568503314727D+15/
DATA B1/-.1025390625000000D+00,.2775764465332031D+00,
& -.1993531733751297D+01,.2724882731126854D+02,
& -.6038440767050702D+03,.1971837591223663D+05,
& -.8902978767070678D+06,.5310411010968522D+08,
& -.4043620325107754D+10,.3827011346598605D+12,
& -.4406481417852278D+14,.6065091351222699D+16/
K0=12
IF (X.GE.35.0) K0=10
IF (X.GE.50.0) K0=8
T1=X-0.25D0*PI
P0=1.0D0
Q0=-0.125D0/X
DO 45 K=1,K0
P0=P0+A(K)*X**(-2*K)
45 Q0=Q0+B(K)*X**(-2*K-1)
CU=DSQRT(RP2/X)
BJ0=CU*(P0*DCOS(T1)-Q0*DSIN(T1))
BY0=CU*(P0*DSIN(T1)+Q0*DCOS(T1))
T2=X-0.75D0*PI
P1=1.0D0
Q1=0.375D0/X
DO 50 K=1,K0
P1=P1+A1(K)*X**(-2*K)
50 Q1=Q1+B1(K)*X**(-2*K-1)
CU=DSQRT(RP2/X)
BJ1=CU*(P1*DCOS(T2)-Q1*DSIN(T2))
BY1=CU*(P1*DSIN(T2)+Q1*DCOS(T2))
ENDIF
DJ0=-BJ1
DJ1=BJ0-BJ1/X
DY0=-BY1
DY1=BY0-BY1/X
RETURN
END
|
