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CHANNEL.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME CHANNEL
* Problem :
* *********
* Analyse the flow of a fluid during injection into a long vertical channel,
* assuming that the flow is modelled by the boundary-value problem
* u'''' = R (u'u'' - u u''') t in [0,1]
* u(0) = 0, u(1) = 1, u'(0) = 0 = u'(1)
* where u is the potential function, u' is the tangential velocity of
* the field, and R is the Reynold's number
* This is problem 7 in the COPS (Version 2) collection of
* E. Dolan and J. More'
* see "Benchmarking Optimization Software with COPS"
* Argonne National Labs Technical Report ANL/MCS-246 (2000)
* SIF input: Nick Gould, November 2000
* classification NOR2-AN-V-V
* The order of the differential equation
IE ND 4 $-PARAMETER
* The number of subintervals
*IE NH 50 $-PARAMETER
*IE NH 100 $-PARAMETER
*IE NH 200 $-PARAMETER
IE NH 400 $-PARAMETER
* The number of collocation points
*IE NC 1 $-PARAMETER
*IE NC 2 $-PARAMETER
*IE NC 3 $-PARAMETER
IE NC 4 $-PARAMETER
* roots of NC-th degree Legendre polynomial
*RE RHO1 0.5 $-PARAMETER for NC = 1
*RE RHO1 0.7886751346 $-PARAMETER for NC = 2
*RE RHO1 0.5 $-PARAMETER for NC = 3
RE RHO1 0.0694318442 $-PARAMETER for NC = 4
*RE RHO2 0.2113248654 $-PARAMETER for NC = 2
*RE RHO2 0.8872983346 $-PARAMETER for NC = 3
RE RHO2 0.3300094782 $-PARAMETER for NC = 4
*RE RHO3 0.1127016654 $-PARAMETER for NC = 3
RE RHO3 0.6699905218 $-PARAMETER for NC = 4
RE RHO4 0.9305681558 $-PARAMETER for NC = 4
* The ODE is defined in [0,TF]
RE TF 1.0
* The uniform interval length
RI RNH NH
R/ H TF RNH
* The Reynolds number
RE R 10.0
* Other useful values
IE 1 1
IE 2 2
IE 3 3
IE 4 4
RE zero 0.0
IA NH-1 NH -1
I+ NC+ND NC ND
RM -R R -1.0
* Factorials
RE FACT0 1.0
DO I 1 NC+ND
RI RI I
IA I-1 I -1
A* FACT(I) FACT(I-1) RI
ND
RE H**ND 1.0
DO I 1 ND
R* H**ND H**ND H
ND
VARIABLES
* The collocation approximation u is defined by the parameters V and W
DO I 1 NH
DO J 1 ND
X V(I,J)
OD J
DO J 1 NC
X W(I,J)
OD J
* DC[i,j,s] is the (s-1)-th derivative of u at the collocation points
DO J 1 NC
DO S 1 ND
X DC(I,J,S)
ND
GROUPS
N OBJ
* - Dc {i in 1..nh,j in 1..nc,s in 1..nd} +
* sum {k in s..nd} v[i,k]*((rho[j]*h)^(k-s)/fact[k-s]) +
* h^(nd-s+1)* sum {k in 1..nc} w[i,k]*(rho[j]^(k+nd-s)/fact[k+nd-s]) = 0
DO I 1 NH
DO J 1 NC
A* RH RHO(J) H
R= HPOWER H**ND
IE S 1
XE DC(I,J,S) DC(I,J,S) -1.0
RE PROD 1.0
IE IND 0
DO K S ND
A/ COEF PROD FACT(IND)
ZE DC(I,J,S) V(I,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
ZE DC(I,J,S) W(I,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
R/ HPOWER HPOWER H
IE S 2
XE DC(I,J,S) DC(I,J,S) -1.0
RE PROD 1.0
IE IND 0
DO K S ND
A/ COEF PROD FACT(IND)
ZE DC(I,J,S) V(I,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
ZE DC(I,J,S) W(I,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
R/ HPOWER HPOWER H
IE S 3
XE DC(I,J,S) DC(I,J,S) -1.0
RE PROD 1.0
IE IND 0
DO K S ND
A/ COEF PROD FACT(IND)
ZE DC(I,J,S) V(I,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
ZE DC(I,J,S) W(I,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
R/ HPOWER HPOWER H
IE S 4
XE DC(I,J,S) DC(I,J,S) -1.0
RE PROD 1.0
IE IND 0
DO K S ND
A/ COEF PROD FACT(IND)
ZE DC(I,J,S) V(I,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
ZE DC(I,J,S) W(I,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
ND
* sum {k in 1..nd} v[nh,k]*(h^(k-1)/fact[k-1]) + h^nd*
* sum {k in 1..nc} w[nh,k]/fact[k+nd-1] - 1 = 0
R= HPOWER H**ND
IE IND 0
RE PROD 1.0
DO K 1 ND
A/ COEF PROD FACT(IND)
ZE BC3 V(NH,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
ZE BC3 W(NH,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
* sum {k in 2..nd} v[nh,k]*(h^(k-2)/fact[k-2]) + h^(nd-1)*
* sum {k in 1..nc} w[nh,k]/fact[k+nd-2] = 0
IE IND 0
RE PROD 1.0
R/ HPOWER HPOWER RH
DO K 2 ND
A/ COEF PROD FACT(IND)
ZE BC4 V(NH,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
ZE BC4 W(NH,K) COEF
R* PROD PROD RH
IA IND IND 1
OD K
* collocation {i in 1..nh,j in 1..nc}
* sum {k in 1..nc} w[i,k]*(rho[j]^(k-1)/fact[k-1]) +
* R*Dc[i,j,1]*Dc[i,j,4] - R*Dc[i,j,2]*Dc[i,j,3] = 0
DO I 1 NH
DO J 1 NC
IE IND 0
RE PROD 1.0
DO K 1 NC
A/ COEF PROD FACT(IND)
ZE C(I,J) W(I,K) COEF
A* PROD PROD RHO(J)
IA IND IND 1
ND
* continuity {i in 1..nh-1,s in 1..nd}:
* sum {k in s..nd} v[i,k]*(h^(k-s)/fact[k-s]) + h^(nd-s+1)*
* sum {k in 1..nc} w[i,k]/fact[k+nd-s] - v[i+1,s] = 0
DO I 1 NH-1
IA I+1 I 1
DO S 1 ND
XE Y(I+1,S) -1.0
IE IND 0
RE PROD 1.0
DO K S ND
A/ COEF PROD FACT(IND)
ZE Y(I+1,S) V(I,K) COEF
R* PROD PROD H
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
ZE Y(I+1,S) W(I,K) COEF
IA IND IND 1
OD K
ND
CONSTANTS
CHANNEL OBJ -1.0
CHANNEL BC3 1.0
CHANNEL BC4 0.0
BOUNDS
XR CHANNEL 'DEFAULT'
XX CHANNEL V(1,1) 0.0
XX CHANNEL V(1,2) 0.0
START POINT
*X CHANNEL1 'DEFAULT' 0.0
DO I 1 NH
* t[i] = (i-1)*h
RI RI I
RA T RI -1.0
R* T T H
* v[i,1] = t[i]^2*(3 - 2*t[i])
RM DIFF T -2.0
RA DIFF DIFF 3.0
R* V1 T DIFF
R* V1 V1 DIFF
A= V(1) V1
Z CHANNEL V(I,1) V1
* v[i,2] = 6*t[i]*(1 - t[i])
RM DIFF T -1.0
RA DIFF DIFF 1.0
R* V2 T DIFF
RM V2 V2 6.0
A= V(2) V2
Z CHANNEL V(I,2) V2
* v[i,3] = 6*(1.0 - 2*t[i])
RM DIFF T -2.0
RA DIFF DIFF 1.0
RM V3 DIFF 6.0
A= V(3) V3
Z CHANNEL V(I,3) V3
* v[i,4] = -12
RE V4 -12.0
A= V(4) V4
Z CHANNEL V(I,4) V4
* w[i,j] = 0
DO J 1 NC
A= W(J) zero
X CHANNEL W(I,J) 0.0
OD J
* Dc {i in 1..nh,j in 1..nc,s in 1..nd} =
* sum {k in s..nd} v[i,k]*((rho[j]*h)^(k-s)/fact[k-s]) +
* h^(nd-s+1)* sum {k in 1..nc} w[i,k]*(rho[j]^(k+nd-s)/fact[k+nd-s])
DO J 1 NC
A* RH RHO(J) H
R= HPOWER H**ND
IE S 1
RE DCIJS 0.0
RE PROD 1.0
IE IND 0
DO K S ND
A/ COEF PROD FACT(IND)
A* COEF V(K) COEF
R+ DCIJS DCIJS COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
A* COEF W(K) COEF
R+ DCIJS DCIJS COEF
R* PROD PROD RH
IA IND IND 1
OD K
Z CHANNEL DC(I,J,S) DCIJS
R/ HPOWER HPOWER H
IE S 2
RE DCIJS 0.0
RE PROD 1.0
IE IND 0
DO K S ND
A/ COEF PROD FACT(IND)
A* COEF V(K) COEF
R+ DCIJS DCIJS COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
A* COEF W(K) COEF
R+ DCIJS DCIJS COEF
R* PROD PROD RH
IA IND IND 1
OD K
Z CHANNEL DC(I,J,S) DCIJS
R/ HPOWER HPOWER H
IE S 3
RE DCIJS 0.0
RE PROD 1.0
IE IND 0
DO K S ND
A/ COEF PROD FACT(IND)
A* COEF V(K) COEF
R+ DCIJS DCIJS COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
A* COEF W(K) COEF
R+ DCIJS DCIJS COEF
R* PROD PROD RH
IA IND IND 1
OD K
Z CHANNEL DC(I,J,S) DCIJS
R/ HPOWER HPOWER H
IE S 4
RE DCIJS 0.0
RE PROD 1.0
IE IND 0
DO K S ND
A/ COEF PROD FACT(IND)
A* COEF V(K) COEF
R+ DCIJS DCIJS COEF
R* PROD PROD RH
IA IND IND 1
OD K
DO K 1 NC
A/ COEF PROD FACT(IND)
R* COEF COEF HPOWER
A* COEF W(K) COEF
R+ DCIJS DCIJS COEF
R* PROD PROD RH
IA IND IND 1
OD K
Z CHANNEL DC(I,J,S) DCIJS
ND
ELEMENT TYPE
EV PROD X Y
ELEMENT USES
DO I 1 NH
DO J 1 NC
XT P1(I,J) PROD
ZV P1(I,J) X DC(I,J,1)
ZV P1(I,J) Y DC(I,J,4)
XT P2(I,J) PROD
ZV P2(I,J) X DC(I,J,2)
ZV P2(I,J) Y DC(I,J,3)
ND
GROUP USES
* collocation {i in 1..nh,j in 1..nc}
* sum {k in 1..nc} w[i,k]*(rho[j]^(k-1)/fact[k-1]) +
* R*Dc[i,j,1]*Dc[i,j,4] - R*Dc[i,j,2]*Dc[i,j,3] = 0
DO I 1 NH
DO J 1 NC
ZE C(I,J) P1(I,J) R
ZE C(I,J) P2(I,J) -R
ND
OBJECT BOUND
* Solution
*LO SOLUTION 1.0
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS CHANNEL
INDIVIDUALS
T PROD
F X * Y
G X Y
G Y X
H X Y 1.0
ENDATA