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spatial_flat.f90
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spatial_flat.f90
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!==============================================================================
subroutine spatial()
!==============================================================================
!
! Driver for spatial parallel stability solver
!
! This is a VERY old version for a flat-plate
!
! S. Scott Collis
!
!==============================================================================
use stuff
use material
implicit none
integer i, j, k, ix, ier
integer i0, idof, j0, jdof
real vm(ny,ndof,nx)
real eta(ny), y(ny), deta(ny), d2eta(ny)
real D1(ny,ny), D2(ny,ny), Dt1(ny,ny), Dt2(ny,ny)
real A(ny,ndof,ndof), B(ny,ndof,ndof), C(ny,ndof,ndof)
real D(ny,ndof,ndof), G(ny,ndof,ndof)
real Vxx(ny,ndof,ndof), Vxy(ny,ndof,ndof), Vyy(ny,ndof,ndof)
real Vxz(ny,ndof,ndof), Vyz(ny,ndof,ndof), Vzz(ny,ndof,ndof)
real g1vm(ny,ndof), g2vm(ny,ndof), g3vm(ny,ndof)
real g11vm(ny,ndof), g12vm(ny,ndof), g13vm(ny,ndof)
real g21vm(ny,ndof), g22vm(ny,ndof), g23vm(ny,ndof)
real g31vm(ny,ndof), g32vm(ny,ndof), g33vm(ny,ndof)
real gum(ny,nsd,nsd), grhom(ny,nsd), gtm(ny,nsd), gpm(ny,nsd)
real divum(ny), g1divum(ny), g2divum(ny), g3divum(ny)
real rhom(ny), u1m(ny), u2m(ny), u3m(ny), tm(ny), pm(ny)
real rmu(ny), dmu(ny), d2mu(ny)
real rlm(ny), dlm(ny), d2lm(ny)
real con(ny), dcon(ny), d2con(ny)
real g1mu(ny), g2mu(ny), g3mu(ny)
real g1lm(ny), g2lm(ny), g3lm(ny)
real g1con(ny), g2con(ny), g3con(ny)
real g1dmu(ny), g2dmu(ny), g3dmu(ny)
real g1dlm(ny), g2dlm(ny), g3dlm(ny)
real g1dcon(ny), g2dcon(ny), g3dcon(ny)
real S1jj(ny), S2jj(ny), S3jj(ny), S(ny,nsd,nsd)
real fact
complex Dh(ny,ndof,ndof), Ah(ny,ndof,ndof), Bh(ny,ndof,ndof)
complex :: im = (0.0,1.0), scale
complex A0(2*ndof*ny,2*ndof*ny), B0(2*ndof*ny,2*ndof*ny), &
D0(2*ndof*ny,2*ndof*ny)
complex C0(ndof*ny,ndof*ny), C1(ndof*ny,ndof*ny), C2(ndof*ny,ndof*ny)
complex Cinv0(ndof*ny,ndof*ny)
complex evec(2*ndof*ny,2*ndof*ny), alp(2*ndof*ny), bet(2*ndof*ny)
complex omg(2*ndof*ny), cs(2*ndof*ny)
real temp1(2*ndof*ny), temp2(2*ndof*ny)
integer index(2*ndof*ny)
!.... stuff for LAPACK eigensolver
integer info, lwork
complex, allocatable :: work(:)
real, allocatable :: rwork(:)
!==============================================================================
!.... grid
call sgengrid(y, eta, deta, d2eta)
!.... read the mean field
call getmean (vm, y, eta)
! call getmean2(vm, y, eta, g2vm, g22vm)
!.... loop over the streamwise stations
do ix = 1, nx
write (*,*)
write (*,*) 'F O R M I N G E I G E N S Y S T E M'
!.... initialize
G = zero
A = zero
B = zero
C = zero
D = zero
Vxx = zero
Vxy = zero
Vyy = zero
Vxz = zero
Vyz = zero
Vzz = zero
rhom = vm(:,1,ix)
u1m = vm(:,2,ix)
u2m = vm(:,3,ix)
u3m = vm(:,4,ix)
tm = vm(:,5,ix)
pm = one / (gamma * Ma**2) * rhom * tm
!.... Compute the Chebyshev derivative matrices
call chebyd(D1, ny-1) ! use ny-1 since this routine uses 0:ny
D2 = matmul(D1, D1) ! compute the second derivative
Dt1 = D1 ! Derivative operator for temperature
Dt1(ny,:) = zero ! adiabatic wall
Dt2 = matmul(D1, Dt1) ! Second Derivative operator for temperature
!.... Compute derivatives of mean field using the appropriate difference scheme
g1vm = zero
do idof = 1, ndof
g2vm(:,idof) = matmul(D1,vm(:,idof,ix))
end do
g3vm = zero
g11vm = zero
g12vm = zero
g13vm = zero
g21vm = zero
do idof = 1, ndof
g22vm(:,idof) = matmul(D2,vm(:,idof,ix))
end do
g23vm = zero
g31vm = zero
g32vm = zero
g33vm = zero
!.... transform the gradients to physical space
do k = 1, ndof
g22vm(:,k) = g22vm(:,k) * deta**2 + g2vm(:,k) * d2eta
g2vm(:,k) = g2vm(:,k) * deta
end do
!.... write out the mean field and its gradients
open(10,file='rho.out')
do j = 1, ny
write (10,13) y(j), vm(j,1,ix), g2vm(j,1), g22vm(j,1)
end do
close(10)
open(10,file='u.out')
do j = 1, ny
write (10,13) y(j), vm(j,2,ix), g2vm(j,2), g22vm(j,2)
end do
close(10)
open(10,file='v.out')
do j = 1, ny
write (10,13) y(j), vm(j,3,ix), g2vm(j,3), g22vm(j,3)
end do
close(10)
open(10,file='t.out')
do j = 1, ny
write (10,13) y(j), vm(j,5,ix), g2vm(j,5), g22vm(j,5)
end do
close(10)
13 format(4(1pe20.13,1x))
!.... initialize gradient of mean velocity
gum(:,1,1) = g1vm(:,2)
gum(:,1,2) = g2vm(:,2)
gum(:,1,3) = g3vm(:,2)
gum(:,2,1) = g1vm(:,3)
gum(:,2,2) = g2vm(:,3)
gum(:,2,3) = g3vm(:,3)
gum(:,3,1) = g1vm(:,4)
gum(:,3,2) = g2vm(:,4)
gum(:,3,3) = g3vm(:,4)
divum = gum(:,1,1) + gum(:,2,2) + gum(:,3,3)
!.... initialize gradient of rho and T in the mean
grhom(:,1) = g1vm(:,1)
grhom(:,2) = g2vm(:,1)
grhom(:,3) = g3vm(:,1)
gtm(:,1) = g1vm(:,5)
gtm(:,2) = g2vm(:,5)
gtm(:,3) = g3vm(:,5)
fact = one / (gamma * Ma**2)
gpm(:,1) = fact * ( grhom(:,1) * tm + rhom * gtm(:,1) )
gpm(:,2) = fact * ( grhom(:,2) * tm + rhom * gtm(:,2) )
gpm(:,3) = fact * ( grhom(:,3) * tm + rhom * gtm(:,3) )
!.... compute the gradient of the divergence of um
g1divum = g11vm(:,2) + g12vm(:,3)
g2divum = g12vm(:,2) + g22vm(:,3)
g3divum = zero
!.... compute some stuff that is useful for the viscous terms
do i = 1, nsd
do j = 1, nsd
S(:,i,j) = pt5 * ( gum(:,i,j) + gum(:,j,i) )
end do
end do
S1jj = pt5 * ( g11vm(:,2) + g11vm(:,2) + g22vm(:,2) + &
g12vm(:,3) + g33vm(:,2) + g13vm(:,4) )
S2jj = pt5 * ( g11vm(:,3) + g21vm(:,2) + g22vm(:,3) + &
g22vm(:,3) + g33vm(:,3) + g23vm(:,4) )
S3jj = pt5 * ( g11vm(:,4) + g31vm(:,2) + g22vm(:,4) + &
g32vm(:,3) + g33vm(:,4) + g33vm(:,4) )
!.... compute mean material properties
call getmat(tm*te, rmu, rlm, con, dmu, d2mu, &
dlm, d2lm, dcon, d2con)
!.... nondimensionalize
rmu = rmu / rmue
dmu = dmu * Te / rmue
d2mu = d2mu * Te**2 / rmue
con = con / cone
dcon = dcon * Te / cone
d2con = d2con * Te**2 / cone
rlm = rlm / rlme
dlm = dlm * Te / rlme
d2lm = d2lm * Te**2 / rlme
!.... compute gradients of viscosity using chain-rule
g1mu = dmu * gtm(:,1)
g2mu = dmu * gtm(:,2)
g3mu = dmu * gtm(:,3)
g1dmu = d2mu * gtm(:,1)
g2dmu = d2mu * gtm(:,2)
g3dmu = d2mu * gtm(:,3)
!.... compute gradients of conductivity using chain-rule
g1con = dcon * gtm(:,1)
g2con = dcon * gtm(:,2)
g3con = dcon * gtm(:,3)
g1dcon = d2con * gtm(:,1)
g2dcon = d2con * gtm(:,2)
g3dcon = d2con * gtm(:,3)
!.... compute gradients of bulk viscosity using chain-rule
g1lm = dlm * gtm(:,1)
g2lm = dlm * gtm(:,2)
g3lm = dlm * gtm(:,3)
g1dlm = d2lm * gtm(:,1)
g2dlm = d2lm * gtm(:,2)
g3dlm = d2lm * gtm(:,3)
!==============================================================================
!.... Continuity equation
G(:,1,1) = one
A(:,1,1) = u1m
A(:,1,2) = rhom
B(:,1,1) = u2m
B(:,1,3) = rhom
C(:,1,1) = u3m
C(:,1,4) = rhom
D(:,1,1) = divum
D(:,1,2) = grhom(:,1)
D(:,1,3) = grhom(:,2)
D(:,1,4) = grhom(:,3)
!.... Momentum equation -- x_1 (convective + pressure)
G(:,2,2) = rhom
A(:,2,1) = tm/(gamma * Ma**2)
A(:,2,2) = rhom * u1m
A(:,2,5) = rhom/(gamma * Ma**2)
B(:,2,2) = rhom * u2m
C(:,2,2) = rhom * u3m
D(:,2,1) = u1m * gum(:,1,1) + u2m * gum(:,1,2) + u3m * gum(:,1,3) &
+ gtm(:,1) / (gamma * Ma**2)
D(:,2,2) = rhom * gum(:,1,1)
D(:,2,3) = rhom * gum(:,1,2)
D(:,2,4) = rhom * gum(:,1,3)
D(:,2,5) = grhom(:,1) / (gamma * Ma**2)
!.... (viscous lambda)
if (.true.) then
fact = rlme / (rmue * Re)
A(:,2,2) = A(:,2,2) - fact * g1lm
A(:,2,5) = A(:,2,5) - fact * dlm * divum
B(:,2,3) = B(:,2,3) - fact * g1lm
C(:,2,4) = C(:,2,4) - fact * g1lm
D(:,2,5) = D(:,2,5) - fact * ( g1dlm * divum + dlm * g1divum )
Vxx(:,2,2) = fact * rlm
Vxy(:,2,3) = fact * rlm
Vxz(:,2,4) = fact * rlm
end if
!.... (viscous mu)
fact = one / Re
if (.true.) then
A(:,2,2) = A(:,2,2) - fact * two * g1mu
A(:,2,3) = A(:,2,3) - fact * g2mu
A(:,2,4) = A(:,2,4) - fact * g3mu
A(:,2,5) = A(:,2,5) - fact * dmu * two * S(:,1,1)
B(:,2,2) = B(:,2,2) - fact * g2mu
B(:,2,5) = B(:,2,5) - fact * dmu * two * S(:,1,2)
C(:,2,2) = C(:,2,2) - fact * g3mu
C(:,2,5) = C(:,2,5) - fact * dmu * two * S(:,1,3)
D(:,2,5) = D(:,2,5) - fact * two * ( g1dmu * S(:,1,1) + &
g2dmu * S(:,1,2) + g3dmu * S(:,1,3) + &
dmu * S1jj )
end if
Vxx(:,2,2) = Vxx(:,2,2) + fact * two * rmu
Vxy(:,2,3) = Vxy(:,2,3) + fact * rmu
Vyy(:,2,2) = Vyy(:,2,2) + fact * rmu
Vxz(:,2,4) = Vxz(:,2,4) + fact * rmu
Vzz(:,2,2) = Vzz(:,2,2) + fact * rmu
!.... Momentum equation -- x_2 (convective + pressure)
G(:,3,3) = rhom
A(:,3,3) = rhom * u1m
B(:,3,1) = tm/(gamma * Ma**2)
B(:,3,3) = rhom * u2m
B(:,3,5) = rhom/(gamma * Ma**2)
C(:,3,3) = rhom * u3m
D(:,3,1) = u1m * gum(:,2,1) + u2m * gum(:,2,2) + u3m * gum(:,2,3) &
+ gtm(:,2) / (gamma * Ma**2)
D(:,3,2) = rhom * gum(:,2,1)
D(:,3,3) = rhom * gum(:,2,2)
D(:,3,4) = rhom * gum(:,2,3)
D(:,3,5) = grhom(:,2) / (gamma * Ma**2)
!.... (viscous lambda)
if (.true.) then
fact = rlme / (rmue * Re)
A(:,3,2) = A(:,3,2) - fact * g2lm
B(:,3,3) = B(:,3,3) - fact * g2lm
B(:,3,5) = B(:,3,5) - fact * dlm * divum
C(:,3,4) = C(:,3,4) - fact * g2lm
D(:,3,5) = D(:,3,5) - fact * ( g2dlm * divum + dlm * g2divum )
Vxy(:,3,2) = fact * rlm
Vyy(:,3,3) = fact * rlm
Vyz(:,3,4) = fact * rlm
end if
!.... (viscous mu)
fact = one / Re
if (.true.) then
A(:,3,3) = A(:,3,3) - fact * g1mu
A(:,3,5) = A(:,3,5) - fact * dmu * two * S(:,2,1)
B(:,3,2) = B(:,3,2) - fact * g1mu
B(:,3,3) = B(:,3,3) - fact * two * g2mu
B(:,3,4) = B(:,3,4) - fact * g3mu
B(:,3,5) = B(:,3,5) - fact * dmu * two * S(:,2,2)
C(:,3,3) = C(:,3,3) - fact * g3mu
C(:,3,5) = C(:,3,5) - fact * dmu * two * S(:,2,3)
D(:,3,5) = D(:,3,5) - fact * two * ( g1dmu * S(:,2,1) + &
g2dmu * S(:,2,2) + g3dmu * S(:,2,3) + &
dmu * S2jj )
end if
Vxx(:,3,3) = Vxx(:,3,3) + fact * rmu
Vxy(:,3,2) = Vxy(:,3,2) + fact * rmu
Vyy(:,3,3) = Vyy(:,3,3) + fact * two * rmu
Vyz(:,3,4) = Vyz(:,3,4) + fact * rmu
Vzz(:,3,3) = Vzz(:,3,3) + fact * rmu
!.... Momentum equation -- x_3 (convective + pressure)
G(:,4,4) = rhom
A(:,4,4) = rhom * u1m
B(:,4,4) = rhom * u2m
C(:,4,1) = tm/(gamma * Ma**2)
C(:,4,4) = rhom * u3m
C(:,4,5) = rhom/(gamma * Ma**2)
D(:,4,1) = u1m * gum(:,3,1) + u2m * gum(:,3,2) + u3m * gum(:,3,3) &
+ gtm(:,3) / (gamma * Ma**2)
D(:,4,2) = rhom * gum(:,3,1)
D(:,4,3) = rhom * gum(:,3,2)
D(:,4,4) = rhom * gum(:,3,3)
D(:,4,5) = grhom(:,3) / (gamma * Ma**2)
!.... (viscous lambda)
if (.true.) then
fact = rlme / (rmue * Re)
A(:,4,2) = A(:,4,2) - fact * g3lm
B(:,4,3) = B(:,4,3) - fact * g3lm
C(:,4,4) = C(:,4,4) - fact * g3lm
C(:,4,5) = C(:,4,5) - fact * dlm * divum
D(:,4,5) = D(:,4,5) - fact * ( g3dlm * divum + dlm * g3divum )
Vxz(:,4,2) = fact * rlm
Vyz(:,4,3) = fact * rlm
Vzz(:,4,4) = fact * rlm
end if
!.... (viscous mu)
fact = one / Re
if (.true.) then
A(:,4,4) = A(:,4,4) - fact * g1mu
A(:,4,5) = A(:,4,5) - fact * dmu * two * S(:,3,1)
B(:,4,4) = B(:,4,4) - fact * g2mu
B(:,4,5) = B(:,4,5) - fact * dmu * two * S(:,3,2)
C(:,4,2) = C(:,4,2) - fact * g1mu
C(:,4,3) = C(:,4,3) - fact * g2mu
C(:,4,4) = C(:,4,4) - fact * two * g3mu
C(:,4,5) = C(:,4,5) - fact * dmu * two * S(:,3,3)
D(:,4,5) = D(:,4,5) - fact * two * ( g1dmu * S(:,3,1) + &
g2dmu * S(:,3,2) + g3dmu * S(:,3,3) + &
dmu * S3jj )
end if
Vxx(:,4,4) = Vxx(:,4,4) + fact * rmu
Vyy(:,4,4) = Vyy(:,4,4) + fact * rmu
Vxz(:,4,2) = Vxz(:,4,2) + fact * rmu
Vyz(:,4,3) = Vyz(:,4,3) + fact * rmu
Vzz(:,4,4) = Vzz(:,4,4) + fact * two * rmu
!.... Energy equation (Advection + pressure)
G(:,5,1) = -gamma1 * tm / gamma
G(:,5,5) = rhom / gamma
A(:,5,1) = -gamma1 * u1m * tm / gamma
A(:,5,5) = rhom * u1m / gamma
B(:,5,1) = -gamma1 * u2m * tm / gamma
B(:,5,5) = rhom * u2m / gamma
C(:,5,1) = -gamma1 * u3m * tm / gamma
C(:,5,5) = rhom * u3m / gamma
D(:,5,1) = one/gamma * ( u1m * gtm(:,1) + u2m * gtm(:,2) + &
u3m * gtm(:,3) )
D(:,5,2) = rhom * gtm(:,1) - gamma1 * Ma**2 * gpm(:,1)
D(:,5,3) = rhom * gtm(:,2) - gamma1 * Ma**2 * gpm(:,2)
D(:,5,4) = rhom * gtm(:,3) - gamma1 * Ma**2 * gpm(:,3)
D(:,5,5) = -gamma1/gamma * ( u1m * grhom(:,1) + u2m * grhom(:,2) + &
u3m * grhom(:,3) )
!.... diffusion
fact = one / (Pr * Re)
A(:,5,5) = A(:,5,5) - fact * (g1con + dcon * gtm(:,1))
B(:,5,5) = B(:,5,5) - fact * (g2con + dcon * gtm(:,2))
C(:,5,5) = C(:,5,5) - fact * (g3con + dcon * gtm(:,3))
D(:,5,5) = D(:,5,5) - fact * (g1dcon * gtm(:,1) + &
g2dcon * gtm(:,2) + g3dcon * gtm(:,3) + &
dcon * ( g11vm(:,5) + g22vm(:,5) + &
g33vm(:,5) ) )
Vxx(:,5,5) = fact * con
Vyy(:,5,5) = fact * con
Vzz(:,5,5) = fact * con
!.... dissipation (lambda)
if (.true.) then
fact = gamma1 * Ma**2 * rlme / (Re * rmue)
A(:,5,2) = A(:,5,2) - fact * two * rlm * divum
B(:,5,3) = B(:,5,3) - fact * two * rlm * divum
C(:,5,4) = C(:,5,4) - fact * two * rlm * divum
D(:,5,5) = D(:,5,5) - fact * dlm * divum * divum
end if
!.... dissipation (mu)
fact = two * gamma1 * Ma**2 / Re
A(:,5,2) = A(:,5,2) - fact * two * rmu * S(:,1,1)
A(:,5,3) = A(:,5,3) - fact * two * rmu * S(:,2,1)
A(:,5,4) = A(:,5,4) - fact * two * rmu * S(:,3,1)
B(:,5,2) = B(:,5,2) - fact * two * rmu * S(:,1,2)
B(:,5,3) = B(:,5,3) - fact * two * rmu * S(:,2,2)
B(:,5,4) = B(:,5,4) - fact * two * rmu * S(:,3,2)
C(:,5,2) = C(:,5,2) - fact * two * rmu * S(:,1,3)
C(:,5,3) = C(:,5,3) - fact * two * rmu * S(:,2,3)
C(:,5,4) = C(:,5,4) - fact * two * rmu * S(:,3,3)
D(:,5,5) = D(:,5,5) - fact * dmu * ( S(:,1,1)**2 + &
S(:,1,2)**2 + S(:,1,3)**2 + S(:,2,1)**2 + &
S(:,2,2)**2 + S(:,2,3)**2 + S(:,3,1)**2 + &
S(:,3,2)**2 + S(:,3,3)**2)
!==============================================================================
!.... initialize
C0 = zero
C1 = zero
C2 = zero
!.... form the equations
Dh = D + im * beta * C + beta**2 * Vzz
Bh = B - im * beta * Vyz
!.... account for the mapping
do idof = 1, ndof
do jdof = 1, ndof
Bh(:,idof,jdof) = Bh(:,idof,jdof) * deta - Vyy(:,idof,jdof) * d2eta
Vyy(:,idof,jdof) = Vyy(:,idof,jdof) * deta**2
end do
end do
!.... C0
i = 1
i0 = (i-1)*ndof
idof = 1
do jdof = 1, ndof
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
do idof = 1, ndof
do jdof = 1, ndof
do i = 2, ny-1
i0 = (i-1)*ndof
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j) - &
Vyy(i,idof,jdof) * D2(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
end do
end do
i = ny
i0 = (i-1)*ndof
idof = 1 ! density equation
do jdof = 1, ndof
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
i = ny
i0 = (i-1)*ndof
idof = ndof ! energy equation
do jdof = 1, ndof-1
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j) - &
Vyy(i,idof,jdof) * D2(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
jdof = ndof
do j = 1, ny
j0 = (j-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * Dt1(i,j) - &
Vyy(i,idof,jdof) * Dt2(i,j)
end do
j0 = (i-1)*ndof
C0(i0+idof,j0+jdof) = C0(i0+idof,j0+jdof) + Dh(i,idof,jdof)
!.... Add the time term into C0 accounting for BC's
do idof = 1, ndof
C0(idof,idof) = C0(idof,idof) - im * omega
end do
do idof = 1, ndof
do jdof = 1, ndof
do i = 2, ny-1
i0 = (i-1)*ndof
C0(i0+idof,i0+jdof) = C0(i0+idof,i0+jdof) - &
im * omega * G(i,idof,jdof)
end do
end do
end do
i = ny
i0 = (i-1)*ndof
do idof = 1, ndof-1
C0(i0+idof,i0+idof) = C0(i0+idof,i0+idof) - im * omega
end do
i = ny
idof = ndof
i0 = (i-1)*ndof
do jdof = 1, ndof
C0(i0+idof,i0+jdof) = C0(i0+idof,i0+jdof) - im * omega * &
G(i,idof,jdof)
end do
!.... form the equations
Dh = im * A + beta * Vxz
Bh = -im * Vxy
!.... account for the mapping
do idof = 1, ndof
do jdof = 1, ndof
Bh(:,idof,jdof) = Bh(:,idof,jdof) * deta
end do
end do
!.... C1
i = 1
i0 = (i-1)*ndof
idof = 1 ! density equation
do jdof = 1, ndof
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
do idof = 1, ndof
do jdof = 1, ndof
do i = 2, ny-1
i0 = (i-1)*ndof
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
end do
end do
i = ny
i0 = (i-1)*ndof
idof = 1 ! density equation
do jdof = 1, ndof
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
i = ny
i0 = (i-1)*ndof
idof = ndof ! energy equation
do jdof = 1, ndof-1
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * D1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
end do
jdof = ndof
do j = 1, ny
j0 = (j-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + &
Bh(i,idof,jdof) * Dt1(i,j)
end do
j0 = (i-1)*ndof
C1(i0+idof,j0+jdof) = C1(i0+idof,j0+jdof) + Dh(i,idof,jdof)
!.... C2
do idof = 1, ndof
do jdof = 1, ndof
do i = 2, ny-1
i0 = (i-1)*ndof
C2(i0+idof,i0+jdof) = Vxx(i,idof,jdof)
end do
end do
end do
i = ny
i0 = (i-1)*ndof
idof = ndof ! energy equation
do jdof = 1, ndof
C2(i0+idof,i0+jdof) = Vxx(i,idof,jdof)
end do
!.... form the extended system
A0 = zero
B0 = zero
evec = zero
alp = zero
bet = zero
omg = zero
! A0(1:ndof*ny,1:ndof*ny) = C0
call LINCG (ndof*ny, C0, ndof*ny, Cinv0, ndof*ny)
B0(1:ndof*ny,1:ndof*ny) = matmul(Cinv0,-C1)
B0(1:ndof*ny,ndof*ny+1:2*ndof*ny) = matmul(Cinv0,-C2)
! B0(1:ndof*ny,1:ndof*ny) = -C1
! B0(1:ndof*ny,ndof*ny+1:2*ndof*ny) = -C2
!.... put in the identity matrices
do i = ndof*ny+1, 2*ndof*ny
! A0(i,i) = one
B0(i,i-ndof*ny) = one
end do
write (*,*)
write (*,*) 'S O L V I N G E I G E N S Y S T E M'
!
!.... IMSL regular complex eigensolver
!
! call LINCG (ndof*ny, C0, ndof*ny, Z0, ndof*ny)
!
! call MCRCR (2*ndof*ny, 2*ndof*ny, B0, 2*ndof*ny, &
! 2*ndof*ny, 2*ndof*ny, A0, 2*ndof*ny, &
! 2*ndof*ny, 2*ndof*ny, D0, 2*ndof*ny)
!
call EVCCG (2*ndof*ny, B0, 2*ndof*ny, alp, evec, 2*ndof*ny)
where (alp .ne. 0)
alp = 1.0 / alp
elsewhere
alp = zero
end where
!
!.... Lapack generalized complex eigensolver
!
! lwork = 8*2*ndof*ny
! allocate (work(lwork), rwork(lwork), STAT=ier)
! if (ier .ne. 0) then
! write(*,*) 'Error allocating work space'
! call exit(1)
! end if
!
! call CGEGV ( 'N', 'V', 2*ndof*ny, A0, 2*ndof*ny, B0, 2*ndof*ny, &
! alp, bet, evec, 2*ndof*ny, evec, 2*ndof*ny, &
! work, lwork, rwork, info )
!
! write (*,*) 'Info = ', info
!
! deallocate (work, rwork)
!
!.... compute the wave number (spatial)
!
! where (bet .ne. 0)
! alp = alp / bet
! elsewhere
! alp = zero
! end where
!
!.... sort the eigenvalues by the imaginary part
do j = 1, 2*ndof*ny
temp2(j) = AIMAG(alp(j))
index(j) = j
end do
call PIKSR2(2*ndof*ny, temp2, index)
do j = 1, 2*ndof*ny
temp1(j) = REAL(alp(index(j)))
A0(:,j) = evec(:,index(j))
end do
alp = cmplx(temp1,temp2)
evec = A0
!.... end loop on x
end do
!.... compute the phase speed
where (alp .ne. 0)
cs = omega / alp
elsewhere
cs = zero
end where
!.... Scale the eigenvectors in a reasonable way
do j = 1, 2*ndof*ny
scale = zero
! do i = ny*ndof+1, 2*ny*ndof
do i = 1, ny*ndof
if ( abs(evec(i,j)) .gt. abs(scale) ) then
scale = evec(i,j)
end if
end do
if (scale .ne. zero) then
! do i = ny*ndof+1, 2*ny*ndof
do i = 1, ny*ndof
evec(i,j) = evec(i,j) / scale
end do
end if
end do
!.... write out the eigensystem in an unformatted output file
open(unit=77,file='evec.dat',form='unformatted',status='unknown')
write (77) ny, ndof, itype
write (77) omega, alpha, beta, Re, Ma, Pr
write (77) y, eta, deta, d2eta
write (77) alp
write (77) evec
close (77)
!.... output the eigenvalues and eigenfunctions to the terminal
if (.false.) then
do j = 1, 2*ndof*ny
if ( aimag(cs(j)) .gt. zero) then
write (*,25) j, abs(alp(j)), real(alp(j)), aimag(alp(j)), &
real(cs(j)), aimag(cs(j))
else
write (*,20) j, abs(alp(j)), real(alp(j)), aimag(alp(j)), &
real(cs(j)), aimag(cs(j))
end if
end do
100 continue
write (*,30)
30 format(/,'Which eigenfunction ==> ',$)
read (*,*) j
if ( j .lt. 0 .or. j .gt. 2*ndof*ny ) goto 100
if (j .ne. 0) then
open (unit=20,file='space.dat',form='formatted',status='unknown')
do i = 1, ny
! i0 = (i+ny-1)*ndof
i0 = (i-1)*ndof
write (20,50)eta(i), &
real(evec(i0+1,j)), &
aimag(evec(i0+1,j)), &
real(evec(i0+2,j)), &
aimag(evec(i0+2,j)), &
real(evec(i0+3,j)), &
aimag(evec(i0+3,j)), &
real(evec(i0+4,j)), &
aimag(evec(i0+4,j)), &
real(evec(i0+5,j)), &
aimag(evec(i0+5,j))