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
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
|
!----------------------------------------------------------------------
!----------------------------------------------------------------------
! stw : Sharp traveling waves
!----------------------------------------------------------------------
!----------------------------------------------------------------------
! Parameter assignment: ( Components of PAR(.) )
!
! PAR : (1) : homotopy parameter
! (2) (3) : A0, A1, A2 (problem parameters)
! (4) (5) (6) : B0, B1, B2 (problem parameters)
! (7) (8) (9) : C0, C1, C2 (problem parameters)
! (10) : c (wave speed)
! (11) : T (period)
! (12) (13) : eps-0, eps-1 (radii)
! (14) (15) : mu-0 , mu-1 (eigenvalues)
! (16) (17) : v-0(1), v-0(2) (eigenvector)
! (18) (19) : v-1(1), v-1(2) (eigenvector)
!-----------------------------------------------------------------------
!-----------------------------------------------------------------------
SUBROUTINE FUNC(NDIM,U,ICP,PAR,IJAC,F,DFDU,DFDP)
! ---------- ----
IMPLICIT NONE
INTEGER, INTENT(IN) :: NDIM, ICP(*), IJAC
DOUBLE PRECISION, INTENT(IN) :: U(NDIM), PAR(*)
DOUBLE PRECISION, INTENT(OUT) :: F(NDIM)
DOUBLE PRECISION, INTENT(INOUT) :: DFDU(NDIM,NDIM), DFDP(NDIM,*)
DOUBLE PRECISION PERIOD
PERIOD=PAR(11)
CALL FFFF(NDIM,U,ICP,PAR,IJAC,F,DFDU)
F(1)=PERIOD*F(1)
F(2)=PERIOD*F(2)
END SUBROUTINE FUNC
SUBROUTINE FFFF(NDIM,U,ICP,PAR,IJAC,F,DFDU)
! ---------- ----
!
! The reduced system for traveling waves is defined here. A separate subroutine
! is used because the system and the Jacobian are also needed in the subroutines
! BCND and ICND below. The computation should be done with JAC=0. The derivatives
! below are for use in BCND and ICND only.
IMPLICIT NONE
INTEGER, INTENT(IN) :: NDIM, ICP(*), IJAC
DOUBLE PRECISION, INTENT(IN) :: U(NDIM), PAR(*)
DOUBLE PRECISION, INTENT(OUT) :: F(NDIM)
DOUBLE PRECISION, INTENT(INOUT) :: DFDU(NDIM,NDIM)
DOUBLE PRECISION FA,FB,FC,C,DFA,DFB,DFC,P(2:9)
! A0, A1, A2:
P(2)= (1-PAR(1))* 2 + PAR(1)* 2
P(3)= (1-PAR(1))* 0 + PAR(1)* 1
! B0, B1, B2:
P(4)= (1-PAR(1))* 2 + PAR(1)* 0
P(5)= (1-PAR(1))* 0 + PAR(1)* 1
P(6)= (1-PAR(1))* 0 + PAR(1)* 0
! C0, C1, C2:
P(7)= (1-PAR(1))* 0 + PAR(1)* 0
P(8)= (1-PAR(1))* 1 + PAR(1)* 1
P(9)= (1-PAR(1))*(-1) + PAR(1)*(-1)
FA = P(2)*U(1) + P(3)*U(1)**2
FB = P(4) + P(5)*U(1) + P(6)*U(1)**2
FC = P(7) + P(8)*U(1) + P(9)*U(1)**2
C = PAR(10)
F(1)= FA * U(2)
F(2)= -C * U(2) - FB * U(2)**2 - FC
IF(IJAC.EQ.0)RETURN
DFA = P(2) + 2*P(3)*U(1)
DFB = P(5) + 2*P(6)*U(1)
DFC = P(8) + 2*P(9)*U(1)
DFDU(1,1)= DFA*U(2)
DFDU(1,2)= FA
DFDU(2,1)= -DFB*U(2)**2 - DFC
DFDU(2,2)= -C - 2*FB*U(2)
END SUBROUTINE FFFF
SUBROUTINE STPNT(NDIM,U,PAR,T)
! ---------- -----
IMPLICIT NONE
INTEGER, INTENT(IN) :: NDIM
DOUBLE PRECISION, INTENT(INOUT) :: U(NDIM),PAR(*)
DOUBLE PRECISION, INTENT(IN) :: T
DOUBLE PRECISION PERIOD, TSC, E
! homotopy parameter :
PAR(1)= 0
! starting period
PERIOD=100.
! c, T:
PAR(10)= 1
PAR(11)= PERIOD
! eps-0, eps-1:
PAR(12)= 0.5*DSQRT(5.D0)/(1+EXP(0.5*PERIOD))
PAR(13)= PAR(12)
! mu-0, mu-1:
PAR(14)= 1
PAR(15)= -1
! v-0(1), v-0(2):
PAR(16)= 2/DSQRT(5.D0)
PAR(17)= 1/DSQRT(5.D0)
! v-1(1), v-1(2):
PAR(18)= 1
PAR(19)= 0
! Exact solution
TSC=PERIOD*(T-0.5)
E=EXP(TSC)
U(1)=1/(1+E)
U(2)=-0.5*E/(1+E)
END SUBROUTINE STPNT
SUBROUTINE BCND(NDIM,PAR,ICP,NBC,U0,U1,FB,IJAC,DBC)
! ---------- ----
IMPLICIT NONE
INTEGER, INTENT(IN) :: NDIM, ICP(*), NBC, IJAC
DOUBLE PRECISION, INTENT(IN) :: PAR(*), U0(NDIM), U1(NDIM)
DOUBLE PRECISION, INTENT(OUT) :: FB(NBC)
DOUBLE PRECISION, INTENT(INOUT) :: DBC(NBC,*)
! Local
DOUBLE PRECISION V0(2),V1(2),G0(2),G1(2),DGDU0(2,2),DGDU1(2,2)
V0(1)=U0(1) - PAR(12)*PAR(16)
V0(2)=U0(2) - PAR(12)*PAR(17)
V1(1)=U1(1) - PAR(13)*PAR(18)
V1(2)=U1(2) - PAR(13)*PAR(19)
CALL FFFF(NDIM,V0,ICP,PAR,1,G0,DGDU0)
CALL FFFF(NDIM,V1,ICP,PAR,1,G1,DGDU1)
! Define eigenvalues and eigenvectors at t=0:
FB(1)= DGDU0(1,1)*PAR(16) + DGDU0(1,2)*PAR(17) - PAR(14)*PAR(16)
FB(2)= DGDU0(2,1)*PAR(16) + DGDU0(2,2)*PAR(17) - PAR(14)*PAR(17)
! Define eigenvalues and eigenvectors at t=1:
FB(3)= DGDU1(1,1)*PAR(18) + DGDU1(1,2)*PAR(19) - PAR(15)*PAR(18)
FB(4)= DGDU1(2,1)*PAR(18) + DGDU1(2,2)*PAR(19) - PAR(15)*PAR(19)
! Normalize the eigenvectors:
FB(5)= PAR(16)**2 + PAR(17)**2 -1
FB(6)= PAR(18)**2 + PAR(19)**2 -1
! Boundary condition at t=0:
FB(7)= G0(1)
FB(8)= G0(2)
! Boundary condition at t=1:
FB(9)= G1(1)
FB(10)=G1(2)
END SUBROUTINE BCND
SUBROUTINE ICND(NDIM,PAR,ICP,NINT,U,UOLD,UDOT,UPOLD,FI,IJAC,DINT)
! ---------- ----
IMPLICIT NONE
INTEGER, INTENT(IN) :: NDIM, ICP(*), NINT, IJAC
DOUBLE PRECISION, INTENT(IN) :: PAR(*)
DOUBLE PRECISION, INTENT(IN) :: U(NDIM), UOLD(NDIM), UDOT(NDIM), UPOLD(NDIM)
DOUBLE PRECISION, INTENT(OUT) :: FI(NINT)
DOUBLE PRECISION, INTENT(INOUT) :: DINT(NINT,*)
! Local
DOUBLE PRECISION DFDU(2,2),F(2),F0(2),DGDU0(1)
CALL FFFF(NDIM,U ,ICP,PAR,1,F ,DFDU )
CALL FFFF(NDIM,UOLD,ICP,PAR,0,F0,DGDU0)
! The integral phase condition is defined here:
FI(1)= ( F(1) - F0(1) ) * ( DFDU(1,1)*F(1) + DFDU(1,2)*F(2) ) &
+ ( F(2) - F0(2) ) * ( DFDU(2,1)*F(1) + DFDU(2,2)*F(2) )
END SUBROUTINE ICND
SUBROUTINE FOPT
END SUBROUTINE FOPT
SUBROUTINE PVLS
END SUBROUTINE PVLS
|
