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flowchan.gms : Flow in a channel COPS 2.0 #7

Analyze the flow of a fluid during injection into a long vertical
channel, assuming that the flow is modeled by the boundary value
problem

        u''''=R(u'u''-uu''),    0<=t<=1,
        u(0)=0, u(1)=1, u'(0)=u'(1)=0,

where u is the potential function, u' is the tangential velocity of
the fluid, and R is the Reynolds number.

This model is from the COPS benchmarking suite.
See http://www-unix.mcs.anl.gov/~more/cops/.

The number of discretization points can be specified using the command
line parameter --nh. COPS performance tests have been reported for nh
= 50, 100, 200, 400

The model can be solved as an NLP (with a dummy objective) or as a
CNS. Select command line parameter --cns to solve the CNS model.

Dolan, E D, and More, J J, Benchmarking Optimization
Software with COPS. Tech. rep., Mathematics and Computer
Science Division, 2000.

Ascher, U M, Mattheij, R M M, and Russell, R D, Numerical
Solution of Boundary Value Problems for Ordinary
Differential Equations. SIAM, 1995.
References:
Large Model of Type: NLP
$Title Flow in a channel COPS 2.0 #7 (FLOWCHAN,SEQ=235) $ontext Analyze the flow of a fluid during injection into a long vertical channel, assuming that the flow is modeled by the boundary value problem u''''=R(u'u''-uu''), 0<=t<=1, u(0)=0, u(1)=1, u'(0)=u'(1)=0, where u is the potential function, u' is the tangential velocity of the fluid, and R is the Reynolds number. This model is from the COPS benchmarking suite. See http://www-unix.mcs.anl.gov/~more/cops/. The number of discretization points can be specified using the command line parameter --nh. COPS performance tests have been reported for nh = 50, 100, 200, 400 The model can be solved as an NLP (with a dummy objective) or as a CNS. Select command line parameter --cns to solve the CNS model. Dolan, E D, and More, J J, Benchmarking Optimization Software with COPS. Tech. rep., Mathematics and Computer Science Division, 2000. Ascher, U M, Mattheij, R M M, and Russell, R D, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, 1995. $offtext $if set n $set nh %n% $if not set nh $set nh 50 $set nc 4 $set nd 4 Sets nc collocation points /1*%nc%/ nh partition intervals /1*%nh%/ nd order of the differential equation /1*%nd%/ Scalars tf ODEs defined in [0 tf] / 1 / R Reynolds number / 10.0 / bc boundary condition / 0 / h uniform interval length; h = tf/%nh%; Parameter rho(nc) roots of k-th degree Legendre polynomial / 1 0.06943184420297 2 0.33000947820757 3 0.66999052179243 4 0.93056815579703 / * The collocation approximation u is defined by the parameters v and w. * uc[i,j] is u evaluated at the collocation points. * Duc[i,j,s] is the (s-1)-th derivative of u at the collocation points. Variables v(nh,nd) w(nh,nc) Duc(nh,nc,nd); Equations Ducdef(nh,nc,nd) continuity(nh,nd) bc_3 bc_4 continuity(nh,nd) collocation(nh,nc); Alias (nh,i), (nc,j,kj), (nd,s,ks); Ducdef{i,j,s}.. Duc(i,j,s) =e= sum {ks$(ord(ks)-ord(s)>=0), v[i,ks]*power(rho[j]*h,ord(ks)-ord(s))/fact[ord(ks)-ord(s)]} + power(h,%nd%-ord(s)+1)* sum{kj, w[i,kj]*power(rho[j],ord(kj)+%nd%-ord(s))/fact[ord(kj)+%nd%-ord(s)]}; * Boundary conditions v.fx('1','1') = bc; v.fx('1','2') = bc; bc_3.. sum {ks, v['%nh%',ks]*power(h,ord(ks)-1)/fact[ord(ks)-1]} + power(h,%nd%)* sum {kj, w['%nh%',kj]/fact[ord(kj)+%nd%-1]} =e= bc; bc_4.. sum {ks$(ord(ks)-2>=0), v['%nh%',ks]*power(h,ord(ks)-2)/fact[ord(ks)-2]} + power(h,%nd%-1)*sum {kj, w['%nh%',kj]/fact[ord(kj)+%nd%-2]} =e= bc; continuity(i+1,s).. sum {ks$(ord(ks)-ord(s)>=0), v[i,ks]*power(h,ord(ks)-ord(s))/fact[ord(ks)-ord(s)]} + power(h,%nd%-ord(s)+1)*sum {kj, w[i,kj]/fact[ord(kj)+%nd%-ord(s)]} =e= v[i+1,s]; collocation(i,j).. sum {kj, w[i,kj]*power(rho[j],ord(kj)-1)/fact[ord(kj)-1]} =e= R*(Duc[i,j,'2']*Duc[i,j,'3'] - Duc[i,j,'1']*Duc[i,j,'4']); * * initial values * Parameter t(nh) partition; t(i) = (ord(i)-1)*h; v.l[i,'1'] = sqr(t[i])*(3 - 2*t[i]); v.l[i,'2'] = 6*t[i]*(1 - t[i]); v.l[i,'3'] = 6*(1.0 - 2*t[i]); v.l[i,'4'] = -12; Duc.l{i,j,s} = sum {ks$(ord(ks)-ord(s)>=0), v.l[i,ks]*power(rho[j]*h,ord(ks)-ord(s))/fact[ord(ks)-ord(s)]}; $if not set cns $goto nlp model channel /all/; $if set workspace channel.workspace = %workspace%; solve channel using cns; $goto continue $label nlp Variable obj dummy objective; Equation defobj; defobj.. obj =e= 0.0; model channel /all/; $if set workspace channel.workspace = %workspace%; solve channel minimizing obj using nlp; $label continue Parameter uc(nh,nc,nd) summary report; uc(i,j,s) = v.l[i,s] + h*sum {kj, w.l[i,kj]*power(rho[j],ord(kj))/fact[ord(kj)]}; display uc;