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orrwong.f
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c***********************************************************************
c> \file orrwong.f
c> \brief Solves the Orr-Sommerfeld equation for incompressible
c> for a truncated bl profile with slip upper boundary
c> \author S. Scott Collis
c***********************************************************************
program orr_wong
c***********************************************************************
c
c Purpose: This program solves the Orr-Sommerfeld equation using a
c Chebyshev-collocation method for a truncated blasius
c boundary layer profile.
c
c The boundary conditions are basically that of a channel
c with a slip wall at the upper boundary.
c
c Author: S. Scott Collis
c
c Date: 3-14-92
c
c Revision: 1-27-93
c
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim)
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
character*20 filename
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type,filename
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
integer nar, nai, ians
real alphar(100), alphai(100)
real minar, maxar, incar, minai, maxai, incai, dar, dai
complex oldeig, dwda
complex eigenvalue(100,100), eigenvector(0:idim), ctemp
logical print
c***********************************************************************
c
c Setup IMSL workspace
c
#ifdef USE_IMSL
REAL RWKSP(100000)
COMMON /WORKSP/ RWKSP
call IWKIN(100000)
#endif
c
c User input
c
write (*,10)
10 format (/,/,10x,'Solve Orr-Sommerfeld (Collocation)')
write (*,20)
20 format (/,1x,'Enter the number of modes ==> ',$)
read (*,*) n
write (*,30)
30 format (/,1x,'Enter Reynolds number ==> ',$)
read (*,*) Re
write (*,40)
40 format (/,1x,'Enter alpha_r (min,max,inc) ==> ',$)
read (*,*) minar,maxar,incar
write (*,45)
45 format (/,1x,'Enter alpha_i (min,max,inc) ==> ',$)
read (*,*) minai,maxai,incai
write (*,50)
50 format (/,1x,'Enter root filename ==> ',$)
read (*,'(a)') filename
write (*,52)
52 format (/,1x,'Print eigenvectors (1,0) ==> ',$)
read (*,*) ians
if (ians.eq.1) then
print = .true.
else
print = .false.
end if
nar = AINT((maxar-minar)/incar)+1
nai = AINT((maxai-minai)/incai)+1
write (*,55) nar, nai
55 format (/,1x,' nar = ',i5,' nai = ',i5)
do i = 1, nar
alphar(i) = minar + (i-1.)*incar
end do
do i = 1, nai
alphai(i) = minai + (i-1.)*incai
end do
nx = nar
ny = nai
oldeig = cmplx(-999.,-999.)
iloc = index(filename,' ')-1
open (unit=10,file=filename(1:iloc)//'.g',form='unformatted',
. status='unknown')
write (10) nx, ny
100 format(i5,1x,$)
write (10)((alphar(i), i=1,nx), j=1,ny),
. ((alphai(j), i=1,nx), j=1,ny)
110 format (1x,e15.8,$)
close (10)
open (unit=12,file=filename(1:iloc)//'.dat',form='formatted',
. status='unknown')
call MAKE_DERIVATIVES
call INIT_BL_PROFILE
write (*,*)
do j = 1, ny
do i = 1, nx
alpha = cmplx (alphar(i), alphai(j))
call MAKE_MATRIX
if (j.eq.2) then
oldeig = eigenvalue(i,j-1)
else if (j.gt.2) then
c oldeig = eigenvalue(i,j-1)+(eigenvalue(i,j-1)-
c . eigenvalue(i,j-2))/(alphai(j-1)-alphai(j-2))*
c . (alphai(j)-alphai(j-1))
oldeig = eigenvalue(i,j-1)
else
oldeig = cmplx(-9999.,-9999.)
end if
call SOLVE(ctemp, eigenvector, print, oldeig, dwda)
eigenvalue(i,j) = ctemp
write (12,60) real(alpha),aimag(alpha),real(eigenvalue(i,j)),
. aimag(eigenvalue(i,j)),real(dwda),aimag(dwda)
write (*,60) real(alpha),aimag(alpha),real(eigenvalue(i,j)),
. aimag(eigenvalue(i,j)),real(dwda),aimag(dwda)
60 format (1x,6(e17.10,1x))
end do
end do
open (unit=11,file=filename(1:iloc)//'.q',form='unformatted',
. status='unknown')
write (11) nx, ny
write (11) 0.0,0.0,0.0,0.0
write (11) ((1.0 , i=1,nx), j=1,ny),
. ((real(eigenvalue(i,j)) , i=1,nx), j=1,ny),
. ((aimag(eigenvalue(i,j)), i=1,nx), j=1,ny),
. ((1.0 , i=1,nx), j=1,ny)
close (11)
stop
end
C***********************************************************************
subroutine MAKE_DERIVATIVES
C***********************************************************************
C
C Make the required matrices that take derivatives in Chebyshev
C space
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim)
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
character*20 filename
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type,filename
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
integer m, p, LDD, i, j, k
real Identity(0:idim,0:idim), D1hat(0:idim,0:idim)
LDD = idim
call CHEBYD (D1hat, LDD, n)
call CHEBYD (D1 , LDD, n)
c
c We enforce noslip indirectly by setting v' = 0 at -1.
c NOTE THAT D1-D4 SHOULD ONLY BE APPLIED TO THE v FIELD ONLY.
c
do i = 0, n
c D1(0,i) = 0.0
D1(N,i) = 0.0
end do
C
C To get higher derivatives just do matrix multiplication
C
do i = 0, n
do j = 0, n
D2(i,j) = 0.0
do k = 0, n
D2(i,j) = D2(i,j) + D1hat(i,k)*D1(k,j)
end do
end do
end do
c
c At the upper surface we allow the flow to slip by setting v'' = 0
c at 1. NOTE THAT D1-D4 SHOULD ONLY BE APPLIED TO THE v FIELD ONLY.
c
do i = 0, n
D2(0,i) = 0.0
c D2(N,i) = 0.0
end do
do i = 0, n
do j = 0, n
D3(i,j) = 0.0
do k = 0, n
D3(i,j) = D3(i,j) + D1hat(i,k)*D2(k,j)
end do
end do
end do
do i = 0, n
do j = 0, n
D4(i,j) = 0.0
do k = 0, n
D4(i,j) = D4(i,j) + D1hat(i,k)*D3(k,j)
end do
end do
end do
c CALL WRRRN ('D1', N+1, N+1, D1, LDd+1, 0)
c CALL WRRRN ('D2', N+1, N+1, D2, LDd+1, 0)
c CALL WRRRN ('D3', N+1, N+1, D3, LDd+1, 0)
c CALL WRRRN ('D4', N+1, N+1, D4, LDd+1, 0)
return
end
C***********************************************************************
subroutine MAKE_BL_METRICS
C***********************************************************************
C
C Setup the collocation points in the mapped coordinate, eta and in
C chebyshev space, th. Also compute the transformation metrics.
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim)
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
character*20 filename
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type,filename
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
real pi, dth
integer i
pi = ACOS(-1.0)
dth = pi/FLOAT(n)
c
c Make mesh in transformed, eta, and Chebyshev space, th
c
do i = 0, n
th(i) = FLOAT(i)*dth
eta(i) = COS(th(i))
end do
c
c Make transformation metrics
c
do i = 0, n
m1(i) = 2.0/Lmap
m2(i) = 0.0
m3(i) = 0.0
m4(i) = 0.0
if (i.eq.0) then
c(i) = 2.
else
c(i) = 1.
end if
end do
return
end
C***********************************************************************
SUBROUTINE FUNCD(X,F,DF)
C***********************************************************************
REAL X, ETA, GAMMA, ETAOUT
COMMON /map/ ETA
GAMMA = 1.2
ETAOUT = 15.0
F = ETAOUT*(1.-TANH(GAMMA))/2.*(X+1)/
. (1.-TANH(GAMMA/2.*(X+1)))-ETA
A = ETAOUT/2.*(1-TANH(GAMMA))
B = GAMMA/2.
DF = A/2.*(1.+EXP(2.*B*(1.+X)) + 2.*B*EXP(2.*B*(1.+X)) +
. 2.*B*EXP(2.*B*(1.+X))*X)
RETURN
END
C***********************************************************************
subroutine INIT_BL_PROFILE
C***********************************************************************
C
C Setup the initial BL profile
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim)
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
character*20 filename
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type,filename
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
parameter (idim2=3000)
integer i, nbl, LDD
character*20 profile
real junk, us(0:idim2), d2us(0:idim2), pi, xitemp
real xi(0:idim2), d1us(0:idim2), d1max, d2max
real ydat(0:idim2), ymin, ymax, h, ut(0:idim2)
real uspl(0:idim2), d2Uspl(0:idim2), metric, Lmap2
real D1hat(0:idim,0:idim), D2hat(0:idim,0:idim)
character*1 temp
pi = ACOS(-1.0)
LDD = idim
if (.true.) then
write (*,10)
10 format (/,1x,'Read Mean Profile',/)
write (*,20)
20 format (1x,'Enter filename ==> ',$)
read (*,'(a)') profile
open (unit=11,file=profile,status='unknown')
i = 0
read (11,*) nbl
35 format (i5)
nbl = nbl - 1
if (nbl*2.gt.idim2) then
write (*,*) 'ERROR: idim2 is too small'
stop
end if
do i = 0,nbl,1
read (11,*,end=100) ydat(i),Ut(i),junk
end do
close (11)
100 continue
else
nbl = 100
do i = 0, nbl
ydat(i) = -1.0+i*1.0/nbl
Ut(i) = (1.0-ydat(i)**2)
end do
end if
c
c Make a mirror image
c
do i = nbl-1, 0, -1
ydat(2*nbl-i) = 2.*ydat(nbl)-ydat(i)
Ut(2*nbl-i) = Ut(i)
end do
nbl2 = 2*nbl
c
c Interpolate data onto a Chebyshev Grid
c
ymin2 = ydat(0)
ymax2 = ydat(nbl2)
h2 = (ymax2-ymin2)/float(nbl)
Lmap2 = ymax2-ymin2
write (*,11) nbl2
11 format (/,1x,'Nbl = ',i5)
call SPLINE(nbl2+1,ydat,Ut,uspl)
ilen = index(filename,' ')-1
open(unit=8,file=filename(1:ilen)//'.org',form='formatted',
. status='unknown')
do i = 0, nbl2
write (8,40) ydat(i),ut(i),uspl(i)
end do
close (8)
dth = pi/(nbl2)
ilen = index(filename,' ')-1
open(unit=9,file=filename(1:ilen)//'.spl',form='formatted',
. status='unknown')
do i = 0, nbl2
thtemp = i*dth
xi(i) = COS(thtemp)*Lmap2/2. + Lmap2/2. + ymin2
call SPEVAL(nbl2+1,ydat,Ut,uspl,xi(i),us(i))
write (9,40) xi(i), us(i), thtemp
end do
close (9)
write (*,66) nbl2
66 format (/,1x,'nbl = ',i5,' Enter ncut ==> ',$)
read (*,*) ifilter
write (*,77) ifilter
77 format (/,1x,'Low pass filter at n = ',i5)
call COSFT(us,nbl2,1)
do i = ifilter, nbl2
us(i) = 0.0
end do
call DCHEBYSHEVF(us,d1us,nbl2)
call DCHEBYSHEVF(d1us,d2us,nbl2)
c call COSFT(us,nbl2,-1)
c call COSFT(d1us,nbl2,-1)
c call COSFT(d2us,nbl2,-1)
c do i = 0, nbl2
c write (10,50) xi(i),us(i),d1us(i),d2us(i)
c 50 format (1x,4(e12.4,1x))
c end do
c
c Convert to theta grid
c
ymin = ydat(0)
ymax = ydat(nbl)
h = (ymax-ymin)/float(nbl)
Lmap = ymax-ymin
call MAKE_BL_METRICS
do i = 0, n
xitemp = COS(th(i))*Lmap/2. + Lmap/2. + ymin
xitemp2 = (xitemp-(ymin2 + Lmap2/2.))*2./Lmap2
if (xitemp2 .ge. 1.) then
xitemp2 = 1.0
else if (xitemp2 .le. -1.) then
xitemp2 = -1.0
end if
thtemp = ACOS( xitemp2 )
call CHEBYINTF(nbl2,u(i),thtemp,us)
call CHEBYINTF(nbl2,d1u(i),thtemp,d1us)
call CHEBYINTF(nbl2,d2u(i),thtemp,d2us)
d1u(i) = d1u(i)*Lmap/Lmap2
d2u(i) = d2u(i)*(Lmap/Lmap2)**2
end do
c
c Compute the collocation derivatives
c
c call CHEBYD (D1hat, LDD, n)
c do i = 0, n
c do j = 0, n
c D2hat(i,j) = 0.0
c do k = 0, n
c D2hat(i,j) = D2hat(i,j) + D1hat(i,k)*D1hat(k,j)
c end do
c end do
c end do
c do i = 0, n
c d1u(i) = 0.0
c d2u(i) = 0.0
c do k = 0, n
c d1u(i) = d1u(i) + D1hat(i,k)*u(k)
c d2u(i) = d2u(i) + D2hat(i,k)*u(k)
c end do
c end do
ilen = index(filename,' ')-1
open(unit=10,file=filename(1:ilen)//'.vel',form='formatted',
. status='unknown')
do i = 0, n
xitemp = eta(i)*Lmap/2. + Lmap/2. + ymin
write (10,12) xitemp,u(i),d1u(i)*2./Lmap,d2u(i)*(2./Lmap)**2
end do
12 format (1x,4(e16.8,4x))
close (10)
write (*,30)
30 format (/,1x,'Velocity Profile completed...',/)
40 format (1x,5(e12.5,2x))
90 format (1x,i5,1x,e12.5)
return
end
C***********************************************************************
subroutine MAKE_MATRIX
C***********************************************************************
C
C This routine generates the matrices which are combined to make
C the generalized eigenvalue problem.
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim)
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
character*20 filename
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type,filename
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
real identity(0:idim,0:idim)
complex work, ai
do i = 0, n
do j = 0, n
A4(i,j) = 0.0
A3(i,j) = 0.0
A2(i,j) = 0.0
A1(i,j) = 0.0
B2(i,j) = 0.0
B1(i,j) = 0.0
B0(i,j) = 0.0
identity(i,j) = 0.0
end do
identity(i,i) = 1.0
end do
C
C Include the independent variable transformation metrics
C
do i = 0, n
work = m1(i)**4
do j = 0, n
A4(i,j) = work*D4(i,j)
end do
end do
do i = 0, n
work = 6.*m1(i)**2*m2(i)
do j = 0, n
A3(i,j) = work*D3(i,j)
end do
end do
do i = 0, n
work = 3.*m2(i)**2+4.*m1(i)*m3(i)-2.*alpha**2*m1(i)**2-
. cmplx(0.,1.)*alpha*Re*U(i)*m1(i)**2
do j = 0, n
A2(i,j) = work*D2(i,j)
end do
end do
do i = 0, n
work = m4(i)-2.*alpha**2*m2(i)-cmplx(0.,1.)*alpha*Re*U(i)*m2(i)
do j = 0, n
A1(i,j) = work*D1(i,j)
end do
end do
do i = 0, n
work = alpha**4+cmplx(0.,1.)*alpha**3*Re*U(i)+
. cmplx(0.,1.)*alpha*Re*(d2u(i)*m1(i)**2+d1u(i)*m2(i))
do j = 0, n
A0(i,j) = work*identity(i,j)
end do
end do
do i = 0, n
work = -1.0*cmplx(0.,1.)*Re*m1(i)**2
do j = 0, n
B2(i,j) = work*D2(i,j)
end do
end do
do i = 0, n
work = -1.0*cmplx(0.,1.)*Re*m2(i)
do j = 0, n
B1(i,j) = work*D1(i,j)
end do
end do
do i = 0, n
work = cmplx(0.,1.)*alpha**2*Re
do j = 0, n
B0(i,j) = work*identity(i,j)
end do
end do
ai = cmplx(0.,1.)
c Make the derivative (wrt alpha) matrices
do i = 0, n
work = -4.*alpha*m1(i)**2 - ai*Re*U(i)*m1(i)**2
do j = 0, n
dA2(i,j) = work*D2(i,j)
end do
end do
do i = 0, n
work = -4.*alpha*m2(i) - ai*Re*U(i)*m2(i)
do j = 0, n
dA1(i,j) = work*D1(i,j)
end do
end do
do i = 0, n
work = 4.*alpha**3 + 3.*ai*alpha**2*Re*U(i) +
. ai*Re*( d2U(i)*m1(i)**2 + d1U(i)*m2(i) )
do j = 0, n
dA0(i,j) = work*identity(i,j)
end do
end do
do i = 0, n
work = ai*2.*alpha*Re
do j = 0, n
dB0(i,j) = work*identity(i,j)
end do
end do
return
end
C***********************************************************************
function CHECKEIG(N,A,LDA,EVAL,EVEC)
C***********************************************************************
C
C Check an eigenvalue and eigenvector
C
C***********************************************************************
integer N
complex A(LDA,N), EVAL, EVEC(N)
complex X(N), Y(N)
real CHECKEIG
#ifdef USE_IMSL
CALL MUCRV (N, N, A, LDA, N, EVEC, 1, N, X)
#else
CALL ZGEMV ('N', N, N, 1.0, A, LDA, EVEC, 1, 0.0, X, 1)
#endif
CHECKEIG = 0.0
DO I = 1, N
CHECKEIG = CHECKEIG + ABS(X(I)-EVAL*EVEC(I))
END DO
CHECKEIG = CHECKEIG/FLOAT(N)
RETURN
END
C***********************************************************************
subroutine HTRAN(N,A,B,LD)
C***********************************************************************
C
C Take the complex conjugate transpose of A and put it in B
C
C***********************************************************************
integer N, LD
complex A(LD,N), B(LD,N)
do i = 1, N
do j = 1, N
B(j,i) = CONJG(A(i,j))
end do
end do
RETURN
END
C***********************************************************************
subroutine CXDOTY (N,X,Y,C)
C***********************************************************************
C
C Take the complex conjugate dot product of vector x and y
C
C***********************************************************************
integer N
complex X(N), Y(N), C
C = CMPLX (0.0,0.0)
do I = 1, N
C = C + CONJG(X(I))*Y(I)
end do
RETURN
END
C***********************************************************************
C S O L V E O R R S O M M E R F E L D
C***********************************************************************
subroutine SOLVE(eigenvalue,eigenvector,print,oldeig,dwda)
C***********************************************************************
C
C This routine generates the discrete eigenvalue problem in
C Chebyshev space for the Orr-Sommerfeld equation.
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=256)
integer n
real eta(0:idim), th(0:idim), lmap
real m1(0:idim), m2(0:idim), m3(0:idim), m4(0:idim)
real c(0:idim), u(0:idim), d1u(0:idim), d2u(0:idim), Re
real D1(0:idim,0:idim), D2(0:idim,0:idim)
real D3(0:idim,0:idim), D4(0:idim,0:idim)
complex A1(0:idim,0:idim), A2(0:idim,0:idim)
complex A3(0:idim,0:idim), A4(0:idim,0:idim)
complex A0(0:idim,0:idim), B0(0:idim,0:idim)
complex B1(0:idim,0:idim), B2(0:idim,0:idim)
complex dA2(0:idim,0:idim), dA1(0:idim,0:idim)
complex dA0(0:idim,0:idim), dB0(0:idim,0:idim)
complex v(0:idim), alpha, omega(0:idim)
character*1 type
character*20 filename
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type,filename
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0, dA2, dA1,
. dA0, dB0
c***********************************************************************
complex A(0:idim,0:idim), B(0:idim,0:idim)
complex dA(0:idim,0:idim), dB(0:idim,0:idim)
complex T1(0:idim,0:idim), T2(0:idim,0:idim)
complex T3(0:idim,0:idim), T4(0:idim,0:idim)
complex eval(0:idim), evec(0:idim,0:idim), tvec(0:idim)
complex aeval(0:idim), aevec(0:idim,0:idim), tavec(0:idim)
complex eigenvalue, eigenvector(0:idim)
complex dwda, dw, dalp, y(0:idim), prod
real temp1(0:idim), temp2(0:idim), residual, CHECKEIG
real temp3(0:idim), temp4(0:idim), dalpha, diff(0:idim)
complex oldeig, ctemp(0:idim)
integer lda, ldb, ldevec, p, i, j, which, k, l, m
integer index(0:idim), index2(0:idim)
logical first, print
external CHECKEIG
C***********************************************************************
lda = idim+1
ldb = idim+1
ldevec = idim+1
do i = 1, n-1
do j = 1, n-1
A(i-1,j-1) = A4(i,j)+A3(i,j)+A2(i,j)+A1(i,j)+A0(i,j)
B(i-1,j-1) = B2(i,j)+B1(i,j)+B0(i,j)
dA(i-1,j-1) = dA2(i,j)+dA1(i,j)+dA0(i,j)
dB(i-1,j-1) = dB0(i,j)
end do
end do
c
c Now enforce the v = 0 @ +_ 1 boundary condition instead of the
c Orr-Sommerfeld equation at n = 0 and n = N. The Neumann
c boundary condition is implicitly enforced.
c
c do j = 0, n
c B(0,j) = CMPLX(0.0, 0.0)
c B(n,j) = CMPLX(0.0, 0.0)
c A(0,j) = CMPLX(0.0, 0.0)
c A(n,j) = CMPLX(0.0, 0.0)
c end do
c A(0,0) = CMPLX(1.0, 0.0)
c A(n,n) = CMPLX(1.0, 0.0)
c
c But the top and bottom rows are trivial so that they can be
c removed
c
#ifdef USE_IMSL
write(*,*) "Solving Eigenvalue Problem with IMSL interface"
CALL LINCG (N-1, B, LDA, T1, LDA)
CALL MCRCR (N-1, N-1, T1, LDA, N-1, N-1, A, LDA,
. N-1, N-1, T4, LDA)
CALL EVCCG (N-1, T4, LDA, eval, evec, ldevec)
#else
write(*,*) "Solving Eigenvalue Problem with new LAPACK interface"
LWORK = 2*(N-1)
INFO = 0
CALL ZLACPY('A', N-1, N-1, B, LDA, T1, LDA)
CALL ZGETRF(N-1, N-1, T1, LDA, IPIV, INFO)
CALL ZGETRI(N-1, T1, LDA, IPIV, WORK, LWORK, INFO)
CALL ZGEMM ('N','N',N-1,N-1,N-1,1.0,T1,LDA,A,LDA,0.0,T4,LDA)
CALL ZGEEV ('N','V',N-1, T4, LDA, eval, VL, LDVL, evec, ldevec,
. WORK, LWORK, RWORK, INFO)
#endif
C
C Solve adjoint problem too
C
#ifdef USE_IMSL
CALL MCRCR (N-1, N-1, A, LDA, N-1, N-1, T1, LDA,
. N-1, N-1, T3, LDA)
call HTRAN (N-1, T3, T2, LDA)
call EVCCG (N-1, T2, LDA, aeval, aevec, ldevec)
#else
write(*,*) "LAPACK Adjoint not yet implemented, use IMSL"
c Need LAPACK here...
#endif
c
do i = 0, N-2
temp1(i) = REAL(eval(i))
temp2(i) = AIMAG(eval(i))
end do
c Need to issolate the most unstable eigenvalue and eigenvector
c Must watch out, this routine isn't very robust.
do i = 0,N-2
index(i) = i
end do
call PIKSR2(n-1,temp2,index)
do i = 0, n-2
temp1(i) = REAL(eval(index(i)))
end do
eigenvalue = cmplx(temp1(n-2),temp2(n-2))
iloc = n-2
c first = .true.
c if ( real(oldeig) .le. -999.) then
c eigenvalue = cmplx(temp1(n-2),temp2(n-2))
c iloc = n-2
c else
c do j = 0, n-2
c diff(j) = abs(real(temp1(j)) - real(oldeig))
c index2(j) = j
c end do
c call PIKSR2(n-1,diff,index2)
c eigenvalue = cmplx(temp1(index2(1)),temp2(index2(1)))
c iloc = index2(1)
c end if
eigenvector(0) = cmplx(0.0,0.0)
eigenvector(n) = cmplx(0.0,0.0)
do j = 1, n-1
eigenvector(j) = evec(j-1,index(iloc))
tvec(j-1) = evec(j-1,index(iloc))
tavec(j-1) = aevec(j-1,index(iloc))
end do
c write (*,120) real(eigenvalue),aimag(eigenvalue),
c . real(oldeig), aimag(oldeig)
c120 format (1x,'Eigs = ',4(e15.8,2x))
residual = CHECKEIG (N-1,T4,lda,eigenvalue,tvec)
if (residual .gt. .01) then
write (*,*) 'WARNING eigenvalue not converged!'
end if
c Compute dw/da
do i = 0, n-2
do j = 0, n-2
T1(i,j) = dB(i,j)*eigenvalue-dA(i,j)
end do
end do
#ifdef USE_IMSL
call MUCRV (N-1, N-1, T1, LDA, N-1, tvec, 1, N-1, temp1)
call CXDOTY (N-1, tavec, temp1, dw)
call MUCRV (N-1, N-1, B, LDA, N-1, tvec, 1, N-1, temp1)
call CXDOTY (N-1, tavec, temp1, dalp)
#else
call ZGEMV ('N', N-1, N-1, 1.0, T1, LDA, tvec, 1, 0.0, ctemp, 1)
call CXDOTY (N-1, tavec, ctemp, dw)
call ZGEMV ('N', N-1, N-1, 1.0, B, LDA, tvec, 1, 0.0, ctemp, 1)
call CXDOTY (N-1, tavec, ctemp, dalp)
#endif
dwda = -dw/dalp
dalpha = -AIMAG(eigenvalue)/AIMAG(dwda)
nalpha = alpha + dalpha
nw = REAL(eigenvalue) + dalpha*REAL(dwda)
c write (*,105) real(dwda), aimag(dwda), dalpha
105 format (/,1x,'dw/da = ',e15.8,', ',e15.8,' i dalpha = ',
. e15.8)
if (print) then
write (*,100) residual
100 format (/,1x,'Residual = ',e17.10)
c write (*,105) real(dwda), aimag(dwda), dalpha
c 105 format (/,1x,'dw/da = ',e15.8,', ',e15.8,' i dalpha = ',
c . e15.8)
c write (*,115) nalpha, nw
c 115 format (/,1x,'nalpha = ',e15.8,', nw = ',e15.8)
write (*,31) Re, real(alpha), aimag(alpha)
write (*,32)
do i = n-2,0,-1
write (*,37) REAL(eval(index(i))/alpha),
. AIMAG(eval(index(i))/alpha),index(i)
c write (20,37) REAL(eval(index(i))),AIMAG(eval(index(i))),
c . REAL(aeval(index(i))),AIMAG(aeval(index(i))),
c . index(i)
end do
write (*,40)
read (*,*) which
do while (which .ne. -1)
temp1(0) = 0.0
temp2(0) = 0.0
temp1(n) = 0.0
temp2(n) = 0.0
temp3(0) = 0.0
temp4(0) = 0.0
temp3(n) = 0.0
temp4(n) = 0.0
do j = 1, n-1
temp1(j) = REAL( evec(j-1,which))
temp2(j) = AIMAG( evec(j-1,which))
temp3(j) = REAL(aevec(j-1,which))
temp4(j) = AIMAG(aevec(j-1,which))
end do
c do j = 0, n
c write (*,38) eta(j),temp1(j),temp2(j),temp3(j),temp4(j)
c end do
c
c Write out normal solution
c
write (*,*)
call WRITEOUT (alpha,eval(which)/alpha,Re,n,temp1,temp2,
. 256,Lmap,ymin)
c
c Write out Adjoint solution
c
c write (*,*)
c call CHEBYINT (n, temp3, temp4, 128)
write (*,40)
read (*,*) which
end do
end if
31 format(/,1x,3(e17.10,4x))
32 format(/,1x,' c_r c_i count',/)
35 format(1x,2(e17.10,4x),i5)
36 format(1x,2(e17.10,4x))
37 format(1x,2(e17.10,2x),i5)
38 format(1x,5(e15.8,1x))
40 format(/,1x,'Eigenvector for which eigenvalue (-1 quits) ==> ',$)
55 format(/,1x,'Performance Index = ',g12.5,/)
return
end
C***********************************************************************
SUBROUTINE WRITEOUT(ALPHA,C,RE,NMODE,EIGR,EIGI,NBIG,LMAP,YMIN)
C***********************************************************************
C
C INTERPOLATE EIGENFUNCTIONS TO A FINER MESH AND OUTPUT
C
c***********************************************************************
INTEGER NBIG, NMODE
REAL EIGR(0:NMODE), EIGI(0:NMODE), LMAP, XI, YMIN
REAL TR, TI, X(0:NBIG), Y(0:NBIG), PI
REAL D1(0:NBIG,0:NBIG),TIME,U(0:NBIG,0:NBIG),V(0:NBIG,0:NBIG)
REAL PSIR(0:NBIG,0:NBIG), PSII(0:NBIG,0:NBIG)