1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
|
#include "cctk.h"
#include "cctk_parameters.h"
#include "cctk_arguments.h"
#include "../../../CactusEinstein/Einstein/src/Einstein.h"
subroutine setupbrilldata3D(CCTK_FARGUMENTS)
c Author: Miguel Alcubierre, October 1998.
c
c Set up 3D Brill data for elliptic solve. The elliptic
c equation we need to solve is:
c
c __
c \/ psi = psi R / 8
c c c
c
c where:
c __
c \/ = Laplacian operator for conformal metric.
c c
c
c R = Ricci scalar for conformal metric.
c c
c
c The Ricci scalar for the conformal metric turns out to be:
c
c / -2q 2 2 -2 2 2 \
c R = 2 | e ( d q + d q ) + rho ( 3 (d q) + 2 d q ) |
c c \ z rho phi phi /
implicit none
DECLARE_CCTK_FARGUMENTS
DECLARE_CCTK_PARAMETERS
integer CCTK_Equals
integer i,j,k
integer nx,ny,nz
CCTK_REAL x1,y1,z1,rho1,rho2
CCTK_REAL phi,phif,e2q
CCTK_REAL brillq,rhofudge,eps
CCTK_REAL zero,one,two,three
c Set up grid size.
nx = cctk_lsh(1)
ny = cctk_lsh(2)
nz = cctk_lsh(3)
c Define numbers
zero = 0.0D0
one = 1.0D0
two = 2.0D0
three = 3.0D0
c Parameters.
rhofudge = brill_rhofudge
eps = brill_eps
c Initialize psi.
brillpsi = one
c Set up conformal metric.
psi = one
do k=1,nz
do j=1,ny
do i=1,nx
x1 = x(i,j,k)
y1 = y(i,j,k)
z1 = z(i,j,k)
rho2 = x1**2 + y1**2
rho1 = dsqrt(rho2)
phi = phif(x1,y1)
e2q = dexp(two*brillq(rho1,z1,phi))
if (rho1.gt.rhofudge) then
gxx(i,j,k) = e2q + (one - e2q)*y1**2/rho2
gyy(i,j,k) = e2q + (one - e2q)*x1**2/rho2
gzz(i,j,k) = e2q
gxy(i,j,k) = - (one - e2q)*x1*y1/rho2
else
gxx(i,j,k) = one
gyy(i,j,k) = one
gzz(i,j,k) = one
gxy(i,j,k) = zero
end if
end do
end do
end do
gxz = zero
gyz = zero
c Set up coefficient Mlinear = - (1/8) Rc.
do k=1,nz
do j=1,ny
do i=1,nx
x1 = x(i,j,k)
y1 = y(i,j,k)
z1 = z(i,j,k)
rho2 = x1**2 + y1**2
rho1 = dsqrt(rho2)
phi = phif(x1,y1)
e2q = dexp(two*brillq(rho1,z1,phi))
c Find M using centered differences over a small
c interval.
if (rho1.gt.rhofudge) then
Mlinear(i,j,k) = - 0.25D0/e2q
. *(brillq(rho1,z1+eps,phi)
. + brillq(rho1,z1-eps,phi)
. + brillq(rho1+eps,z1,phi)
. + brillq(rho1-eps,z1,phi)
. - 4.0D0*brillq(rho1,z1,phi))
. / eps**2
else
Mlinear(i,j,k) = - 0.25D0/e2q
. *(brillq(rho1,z1+eps,phi)
. + brillq(rho1,z1-eps,phi)
. + two*brillq(eps,z1,phi)
. - two*brillq(rho1,z1,phi))
. / eps**2
end if
c Here I assume that for very small rho, the
c phi derivatives are essentially zero. This
c must always be true otherwise the function
c is not regular on the axis.
if (rho1.gt.rhofudge) then
Mlinear(i,j,k) = Mlinear(i,j,k) - 0.25D0/rho2
. *(three*0.25D0*(brillq(rho1,z1,phi+eps)
. - brillq(rho1,z1,phi-eps))**2
. + two*(brillq(rho1,z1,phi+eps)
. - two*brillq(rho1,z1,phi)
. + brillq(rho1,z1,phi-eps)))/eps**2
end if
end do
end do
end do
c Set coefficient Nsource = 0.
Nsource = zero
return
end
|
