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orrfdchan.f
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orrfdchan.f
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c***********************************************************************
c> \file orrfdchan.f
c> \brief Solves the Orr-Sommerfeld equation for incompressible planar
c> channel flow using finite-differences
c> \author S. Scott Collis
c***********************************************************************
program orr_fd_chan
c***********************************************************************
c
c Purpose: This program solves the Orr-Sommerfeld equation using a
c finite-difference method for channel profiles.
c
c Author: S. Scott Collis
c
c Date: 2-29-92
c
c Revision: 9-18-92
c
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=512)
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 v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0
c***********************************************************************
integer nar, nai, ians
real alphar(100), alphai(100)
real minar, maxar, incar, minai, maxai, incai, dar, dai
complex eigenvalue(100,100), eigenvector(0:idim), ctemp
character*9 filename
logical print
c***********************************************************************
c
c Setup IMSL workspace
c
#ifdef USE_IMSL
REAL RWKSP(150000)
COMMON /WORKSP/ RWKSP
call IWKIN(150000)
#endif
c
c User input
c
write (*,10)
10 format (/,/,10x,'Solve Orr-Sommerfeld (Finite-difference)')
write (*,20)
20 format (/,1x,'Enter the number of mesh points ==> ',$)
read (*,*) n
write (*,30)
30 format (/,1x,'Enter Reynolds number ==> ',$)
read (*,*) Re
c write (*,65)
c 65 format (/,1x,'Enter mapping metric ==> ',$)
c read (*,*) Lmap
lmap = 1.0
write (*,40)
40 format (/,1x,'Enter alpha_r (min,max,inc) ==> ',$)
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
iloc = index(filename,' ')-1
open (unit=10,file=filename(1:iloc)//'.g',form='formatted',
. status='unknown')
write (10,100) nx, ny
100 format(5(i5,1x))
write (10,110) ((alphar(i), i=1,nx), j=1,ny),
. ((alphai(j), i=1,nx), j=1,ny)
110 format (5(ES16.8E3,1x))
close (10)
open (unit=12,file=filename(1:iloc)//'.dat',form='formatted',
. status='unknown')
call MAKE_CHANNEL_METRICS
call MAKE_DERIVATIVES
call INIT_CHANNEL_PROFILE
do j = 1, ny
do i = 1, nx
alpha = cmplx (alphar(i), alphai(j))
call MAKE_MATRIX
call SOLVE_ORR_SOM(ctemp, eigenvector, print)
eigenvalue(i,j) = ctemp
write (12,60) real(alpha),aimag(alpha),real(eigenvalue(i,j)),
. aimag(eigenvalue(i,j))
write (*,60) real(alpha),aimag(alpha),real(eigenvalue(i,j)),
. aimag(eigenvalue(i,j))
60 format (1x,4(ES16.8E3,1x))
end do
end do
open (unit=11,file=filename(1:iloc)//'.q',form='formatted',
. status='unknown')
write (11,100) nx, ny
write (11,110) 0.0,0.0,0.0,0.0
write (11,110) ((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_CHANNEL_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=512)
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 v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0
c***********************************************************************
real pi, deta
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) = 1.0
m2(i) = 0.
m3(i) = 0.
m4(i) = 0.
if (i.eq.0) then
c(i) = 2.
else
c(i) = 1.
end if
end do
return
end
C***********************************************************************
subroutine INIT_CHANNEL_PROFILE
C***********************************************************************
C
C Setup the initial channel profile
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=512)
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 v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0
c***********************************************************************
integer i, LDD
real D1hat(0:idim,0:idim), D2hat(0:idim,0:idim)
LDD = idim
do i = 0, n
u(i) = (1.-eta(i)**2)
d1u(i) = -2.0*eta(i)
d2u(i) = -2.0
end do
#ifdef USE_FD_DERIVATIVES
c
c Compute the finite difference derivatives
c
call FiniteD1 (D1hat, LDD, n, eta)
do i = 0, n
do j = 0, n
D2hat(i,j) = 0.0
do k = 0, n
D2hat(i,j) = D2hat(i,j) + D1hat(i,k)*D1hat(k,j)
end do
end do
end do
do i = 0, n
d1u(i) = 0.0
d2u(i) = 0.0
do k = 0, n
d1u(i) = d1u(i) + D1hat(i,k)*u(k)
d2u(i) = d2u(i) + D2hat(i,k)*u(k)
end do
end do
#endif
open(10)
do i = 0, n
write (10,10) eta(i), u(i), d1u(i), d2u(i)
end do
10 format (1x,4(ES16.8E3,4x))
close(10)
return
end
C***********************************************************************
subroutine MAKE_DERIVATIVES
C***********************************************************************
C
C Make the required matrices that take derivatives in Finite diff
C space
C
c***********************************************************************
c Common variables
c***********************************************************************
parameter (idim=512)
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 v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0
c***********************************************************************
integer m, p, LDD, i, j, k
real Identity(0:idim,0:idim), D1hat(0:idim,0:idim)
LDD = idim
call FiniteD1 (D1hat, LDD, n, eta)
call FiniteD1 (D1 , LDD, n, eta)
c
c We enforce the Neumann boundary conditions indirectly by setting
c v' = 0 at 0 and n. NOTE THAT D1-D4 SHOULD ONLY BE APPLIED TO THE
c v FIELD ONLY.
c
do i = 0, n
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
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=512)
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 v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0
c***********************************************************************
real pi, deta
integer i
pi = ACOS(-1.0)
deta = 2./float(n)
c
c Make mesh in transformed, eta
c
do i = 0, n
eta(i) = 1. - float(i)*deta
end do
c
c Make transformation metrics
c
do i = 0, n
m1(i) = (eta(i)-1.)**2/(2.*Lmap)
m2(i) = (eta(i)-1.)**3/(2.*Lmap**2)
m3(i) = 3.*(eta(i)-1.)**4/(4.*Lmap**3)
m4(i) = 3.*(eta(i)-1.)**5/(2.*Lmap**4)
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=512)
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 v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0
c***********************************************************************
parameter (nu = 64)
real utemp(0:nu),utemp1(0:nu),utemp2(0:nu),junk, pi
real gamma, yout, y, x, xi, IU, IV, IW
integer nmode, i, merge
character*15 filename
integer LDD
real D1hat(0:idim,0:idim), D2hat(0:idim,0:idim)
real RTNEWT
external RTNEWT
common /map/ x
pi = ACOS(-1.0)
LDD = idim
gamma = 1.2
xout = 15.0
write (*,5)
5 format (/,1x,'Read Mean Profile',/)
write (*,9)
9 format (1x,'Enter filename ==> ',$)
read (*,'(a)') filename
open (unit=11,file=filename,status='unknown')
read (11,*) nmode
do i = 0, nmode
read (11,*) y,utemp(i),junk
end do
close (11)
c
c I need to spectrally interpolate this profile onto the new grid.
c
call CHEBYSHEV (utemp,nmode,1)
c
c y is nondimensionalized by dr = sqrt(2*x*nu/u_inf)
c
i = n
y = Lmap*(1.+eta(i))/(1.-eta(i))
do while (y/sqrt(2.) .le. xout)
u(i) = 0.0
X = Y/SQRT(2.)
xi = RTNEWT(-2.,2.2,1e-12,FUNCD)
do m = 0, nmode
u(i) = u(i)+utemp(m)*COS(float(m)*ACOS(xi))
end do
i = i - 1
y = Lmap*(1.+eta(i))/(1.-eta(i))
end do
merge = i
write (*,11) merge
11 format(/,1x,'Merging at i = ',i4)
do j = merge, 0, -1
u(j) = 1.0
end do
c
c Compute the collocation derivatives
c
call FiniteD1 (D1hat, LDD, n, eta)
do i = 0, n
do j = 0, n
D2hat(i,j) = 0.0
do k = 0, n
D2hat(i,j) = D2hat(i,j) + D1hat(i,k)*D1hat(k,j)
end do
end do
end do
do i = 0, n
d1u(i) = 0.0
d2u(i) = 0.0
do k = 0, n
d1u(i) = d1u(i) + D1hat(i,k)*u(k)
d2u(i) = d2u(i) + D2hat(i,k)*u(k)
end do
end do
c
c Ensure that the derivatives are zero when they are supposed to be.
c
do j = merge-2, 0, -1
d1u(j) = 0.0
d2u(j) = 0.0
end do
c
c Write out mean profile
c
open(10)
do i = 0, n
write (10,10) eta(i), u(i), d1u(i), d2u(i)
end do
10 format (1x,4(ES16.8E3,1x))
close(10)
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=512)
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 v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0
c***********************************************************************
real rwork(0:idim), cwork(0:idim), identity(0:idim,0:idim)
complex work
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
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 SOLVE_ORR_SOM(eigenvalue, eigenvector, print)
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=512)
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 v(0:idim), alpha, omega(0:idim)
character*1 type
common /data/ n,eta,th,m1,m2,m3,m4,c,u,d1u,d2u,Re,v,
. alpha,omega,type
common /deriv/ D1,D2,D3,D4,lmap
common /matrix/ A4, A3, A2, A1, A0, B2, B1, B0
c***********************************************************************
complex A(0:idim,0:idim), B(0:idim,0:idim), alp(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 beta(0:idim), evec(0:idim,0:idim), temp3(0:idim)
complex evec2(0:idim,0:idim), eval(0:idim)
complex avec(0:idim,0:idim)
complex P1(0:idim,0:idim), P2(0:idim,0:idim)
complex eigenvalue, eigenvector(0:idim)
real temp1(0:idim), temp2(0:idim), residual
integer lda, ldb, ldevec, ldavec, p, i, j, which, k, l, m
real rind(0:idim)
integer ind(0:idim)
logical first, print
external CHECKEIG
complex escale
integer INFO, LWORK, IPIV(0:idim)
complex WORK(16*(idim+1))
real RWORK(8*(idim+1))
C***********************************************************************
LDA = idim+1
LDB = idim+1
LDEVEC = idim+1
LDAVEC = 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)
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
c 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
#ifdef OS_USE_GENERALIZED_EIGENSOLVER
c
c.... use generalized eigenvalue solver
c
LWORK = 16*(idim+1)
INFO = 0
call ZGEGV( 'V', 'V', N-1, A, LDA, B, LDA, alp, beta, avec,
. LDA, evec, LDA, work, lwork, rwork, info)
c
c.... compute the eigenvalues (given generalized eigenvalues)
c
do i = 0, N-2
if (beta(i).ne.0) then
eval(i) = alp(i)/beta(i)
else
eval(i) = 0.0
end if
end do
#else
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('V','V',N-1, T4, LDA, eval, avec, LDAVEC,
& evec, LDEVEC, WORK, LWORK, RWORK, INFO)
#endif
#endif
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
rind(i) = REAL(i)
end do
call SORT(n-1,temp2,rind)
do i = 0,N-2
ind(i) = INT(rind(i))
end do
first = .true.
do i = n-2,0,-1
if ((abs(temp2(i)).lt. 10.0).and.(first).and.
& (int(temp1(ind(i))).ne.999)) then
first = .false.
eigenvalue = cmplx(temp1(ind(i)),temp2(i))
iloc = i
end if
end do
#ifdef ORR_CHECK_EIG
do j = 0, n-2
eigenvector(j) = evec(j,ind(iloc))
end do
residual = CHECKEIG(N-1,T4,lda,eigenvalue,eigenvector)
if (residual .gt. .01) then
write (*,*) 'WARNING eigenvalue not converged!'
end if
#endif
if (print) then
write (*,100) residual
100 format (/,1x,'Residual = ',f17.10)
write (*,31) Re, real(alpha), aimag(alpha)
write (*,32)
do i = 1, N
write (*,35) REAL(eval(i)), AIMAG(eval(i)), i
write (20,36) REAL(eval(i))/alpha, AIMAG(eval(i))/alpha
end do
write (*,40)
read (*,*) which
do while (which .ne. 0)
#if 1
c
c.... normalize the eigenvector for output
c
escale = 0.0
do i = 1, N-1
if (abs(evec(i-1,ind(iloc))) .gt. abs(escale)) then
escale = evec(i-1,ind(iloc))
endif
end do
write(*,*) "escale = ", escale
escale = 1.0/(escale)
call zscal( N-1, escale, evec(0,ind(iloc)), 1)
#endif
temp1(0) = 0.0
temp2(0) = 0.0
temp1(n) = 0.0
temp2(n) = 0.0
do j = 1, n-1
temp1(j) = REAL(evec(j-1,ind(iloc)))
temp2(j) = AIMAG(evec(j-1,ind(iloc)))
end do
open(11)
do j = 0, N
write (11,60) eta(j), temp1(j), temp2(j),
& REAL(avec(j,ind(iloc))), AIMAG(avec(j,ind(iloc)))
end do
close(11)
write (*,40)
read (*,*) which
end do
end if
31 format(/,1x,3(e17.10,4x))
32 format(/,1x,' w_r w_i count',/)
35 format(1x,2(e17.10,4x),i5)
36 format(1x,2(e17.10,4x))
40 format(/,1x,'Eigenvector for which eigenvalue (0 quits) ==> ',$)
55 format(/,1x,'Performance Index = ',g12.5,/)
60 format(1x,6(ES16.8E3,1x))
return
end
C***********************************************************************
SUBROUTINE DCHEBYSHEV(N, Y, DY)
C***********************************************************************
C
C Calculate the Chebyshev transform of a set Y of N+1 real-valued
C data points and compute the derivative in Chebyshev space.
C Then inverse transform and return the derivative in DY in real
C space. Note that Y is returned unscathed.
C
C***********************************************************************
INTEGER N
REAL Y(0:N), DY(0:N)
PARAMETER (idim=512)
REAL WORK(0:IDIM)
IF (N.GT.IDIM) THEN
WRITE (*,*) 'ERROR. N > IDIM in DCHEBYSHEV'
STOP
END IF
C
C SAVE THE INPUT VECTOR
C
DO I = 0, N
WORK(I) = Y(I)
END DO
C
C COMPUTE THE CHEBYSHEV TRANSFORM
C
CALL CHEBYSHEV(Y,N,1)
C
C NOW USE THE RECURSIVE RELATION TO TAKE THE DERIVATIVE
C
DY(N) = 0.0
DY(N-1) = 2.*FLOAT(N)*Y(N)
DO K = N-2, 0, -1
DY(K) = DY(K+2) + 2.*(K+1.)*Y(K+1)
END DO
DY(0) = DY(0)/2.
C
C INVERSE TRANSFORM TO GET BACK TO REAL SPACE
C
CALL CHEBYSHEV(DY,N,-1)
DO I = 0, N
Y(I) = WORK(I)
END DO
RETURN
END
C***********************************************************************
SUBROUTINE ICHEBYSHEV(Y,YI,N)
C***********************************************************************
C
C Calculate the Chebyshev transform of a set Y of N+1 real-valued
C data points and integrate in Chebyshev space. The integral is
C returned in real space in YI and Y is unscathed.
C
C***********************************************************************
REAL Y(0:N), YI(0:N+1)
PARAMETER (idim=512)
REAL WORK(0:IDIM)
DO I = 0, N
WORK(I) = Y(I)
END DO
C
C COMPUTE THE CHEBYSHEV TRANSFORM
C
CALL CHEBYSHEV(Y,N,1)
C
C NOW INTEGRATE
C NOTE I AM ASSUMING THAT Y(N) IS SMALL SUCH THAT YI(N+1) = 0.0
C
YI(N+1) = Y(N)/(2.*(N+1.))
YI(N) = Y(N-1)/(2.*N)
YI(1) = 1./2.*(2.*Y(0)-Y(2))
YI(0) = YI(1)
DO K = 2, N-1
YI(K) = 1./(2.*K)*(Y(K-1)-Y(K+1))
YI(0) = YI(0) + (-1.)**(K-1)*YI(K)
END DO
CALL CHEBYSHEV(YI,N,-1)
DO I = 0, N
Y(I) = WORK(I)
END DO
RETURN
END
C***********************************************************************
SUBROUTINE CHEBYINT(nmode,U,V,NBIG)
C***********************************************************************
C
C Interpolate two functions to a finer mesh
C
c***********************************************************************
INTEGER NBIG, nmode
REAL U(0:Nmode), V(0:Nmode)
REAL IU, IV, X, PI
PI = ACOS(-1.0)
call CHEBYSHEV (U,nmode,1)
call CHEBYSHEV (V,nmode,1)
do i = 0, NBIG
X = I*PI/NBIG
IU = 0.0
IV = 0.0