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partition_wind.f90
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module partition_wind
use current_kind, only: cp, Zero, One, Quarter, Four, Half, Eighth, Two, Third
use diagnostics, only: RGas, Grav, Kappa
implicit none
private
public :: partition, balance_lhs, balance_lhs_exp, balance_lhs_tlm, &
balance_lhs_adj, balance_rhs_simp, balance_rhs_simp_exp, balance_rhs_simp_tlm, &
balance_rhs_simp_adj, imbalance_simp_adj, imbalance_exp, pv_exp, pv_tlm, pv_adj, &
solve_phi_psi
contains
! Extents should be as follows:
! us/mu - (0:nx,ny) ; vs/mv - (nx,0:ny) ; div/mchi - (nx,ny)
! vort/mpsi - (nx-1,ny-1) ; chi - (0:nx+1,0:ny+1) ; psi - (0:nx,0:ny)
subroutine partition(us, vs, div, vort, mchi, mpsi, mu, mv, phill, rdx, chi, psi)
real(cp), dimension(0:,:), intent(in) :: us, mu
real(cp), dimension(:,0:), intent(in) :: vs, mv
real(cp), dimension(:,:), intent(in) :: div, vort, mchi, mpsi
real(cp), intent(in) :: phill, rdx
real(cp), dimension(0:size(us,1),0:size(vs,2)), intent(out) :: chi
real(cp), dimension(0:size(vs,1),0:size(us,2)), intent(out) :: psi
real(cp), parameter :: Eps = 1e-4_cp
integer, parameter :: MaxIter = 50000
character(len=*), parameter :: Mismatch = "mismatch in partition!"
real(cp), dimension(:,:,:), allocatable :: ps, ch
real(cp), dimension(size(div,1),size(div,2)) :: divForcing, divCheck
real(cp), dimension(0:size(div,1),size(div,2)) :: uu
real(cp), dimension(size(div,1),0:size(div,2)) :: vv
real(cp), dimension(size(div,1)-1,size(div,2)) :: u
real(cp), dimension(size(div,1),size(div,2)-1) :: v
real(cp), dimension(size(vort,1),size(vort,2)) :: vortForcing, vortCheck
real(cp) :: dx, dx2, err
integer :: nx, ny, prev, cur, i, j, k
nx = size(vs,1)
ny = size(us,2)
dx = One / rdx
dx2 = dx**2
divForcing = dx2 * div / mchi**2
vortForcing = dx2 * vort / mpsi**2
! Sanity checks
if (any(shape(us) /= shape(mu))) stop "us mu " // Mismatch
if (any(shape(vs) /= shape(mv))) stop "vs mv " // Mismatch
if (any(shape(div) /= shape(mchi))) stop "div " // Mismatch
if (any(shape(vort) /= shape(mpsi))) stop "vort " // Mismatch
if (any(shape(mchi) /= (/ nx, ny /))) stop "mchi " // Mismatch
if (any(shape(mpsi) /= (/ nx-1, ny-1 /))) stop "mpsi " // Mismatch
if (any(shape(chi) /= (/ nx+2, ny+2 /))) stop "chi " // Mismatch
if (any(shape(psi) /= (/ nx+1, ny+1 /))) stop "psi " // Mismatch
! Initialize
allocate(ch(0:nx+1,0:ny+1,2), ps(0:nx,0:ny,2))
prev = 1; cur = 2
! First guess
ch = Zero ; ps = Zero
do k = 1, MaxIter
prev = 2 - mod(k,2)
cur = 2 - mod(k+1,2)
! Compute boundary values of psi
! ps(0,1,cur) = (One - Eighth) * ps(0,1,prev) + Eighth * (Two * phill - ps(1,1,prev))
! print *, phill, ps(0,1,cur), ps(1,1,prev)
! do j = 2, ny ! Set ps(0,1) to zero
!!!! do j = 1, ny
! ps(0,j,cur) = ps(1,j,prev) + &
! (ch(1,j+1,prev) - ch(1,j,prev)) - dx * vs(1,j) / mv(1,j)
! end do
ps(0,2:,cur) = ps(1,2:,prev) + &
(ch(1,3:,prev) - ch(1,2:ny,prev)) - dx * vs(1,2:) / mv(1,2:)
!!$ do i = 1, nx
!!$ ps(i,ny,cur) = ps(i,ny-1,prev) + &
!!$ (ch(i+1,ny,prev) - ch(i,ny,prev)) - dx * us(i,ny) / mu(i,ny)
!!$ end do
ps(1:,ny,cur) = ps(1:,ny-1,prev) + &
(ch(2:,ny,prev) - ch(1:nx,ny,prev)) - dx * us(1:,ny) / mu(1:,ny)
!!$ do j = ny-1, 0, -1
!!$ ps(nx,j,cur) = ps(nx-1,j,prev) - &
!!$ (ch(nx,j+1,prev) - ch(nx,j,prev)) + dx * vs(nx,j) / mv(nx,j)
!!$ end do
ps(nx,:ny-1,cur) = ps(nx-1,:ny-1,prev) - &
(ch(nx,1:ny,prev)-ch(nx,:ny-1,prev)) + dx*vs(nx,:ny-1)/mv(nx,:ny-1)
!!$ do i = nx-1, 1, -1
!!$ ps(i,0,cur) = ps(i,1,prev) - &
!!$ (ch(i+1,1,prev) - ch(i,1,prev)) + dx * us(i,1) / mu(i,1)
!!$ end do
ps(1:nx-1,0,cur) = ps(1:nx-1,1,prev) - &
(ch(2:nx,1,prev)-ch(1:nx-1,1,prev)) + dx*us(1:nx-1,1)/mu(1:nx-1,1)
!!$ do j = 1, ny
!!$ ps(0,j,cur) = ps(0,j-1,cur) + &
!!$ (ch(1,j,prev) - ch(0,j,prev)) - dx * us(0,j) / mu(0,j)
!!$ end do
!!$ do i = 1, nx-1
!!$ ps(i,ny,cur) = ps(i-1,ny,cur) - &
!!$ (ch(i,ny+1,prev) - ch(i,ny,prev)) + dx * vs(i,ny) / mv(i,ny)
!!$ end do
!!$ do i = 1, nx
!!$ ps(i,0,cur) = ps(i-1,0,cur) - &
!!$ (ch(i,1,prev) - ch(i,0,prev)) + dx * vs(i,0) / mv(i,0)
!!$ end do
!!$ do j = 1, ny-1
!!$ ps(nx,j,cur) = ps(nx,j-1,cur) + &
!!$ (ch(nx+1,j,prev) - ch(nx,j,prev)) - dx * us(nx,j) / mu(nx,j)
!!$ end do
! Compute interior values of psi
call update_sor(ps, vortForcing, prev, cur)
!!$ do j = 1, ny-2
!!$ do i = 1, nx-2
!!$ ps(i,j,cur) = ps(i,j,prev) + Quarter * Omega * ( &
!!$ ps(i-1,j,cur) + ps(i+1,j,prev) + ps(i,j-1,cur) + &
!!$ ps(i,j+1,prev) - Four * ps(i,j,prev) - &
!!$ dx2 * vort(i,j) / mpsi(i,j)**2 )
!!$ end do
!!$ ! i = nx-1 special
!!$ ps(i,j,cur) = ps(i,j,prev) + Quarter * Omega * ( &
!!$ ps(i-1,j,cur) + ps(i+1,j,cur) + ps(i,j-1,cur) + &
!!$ ps(i,j+1,prev) - Four * ps(i,j,prev) - &
!!$ dx2 * vort(i,j) / mpsi(i,j)**2 )
!!$ end do
!!$ ! j = ny-1 special
!!$ do i = 1, nx-2
!!$ ps(i,j,cur) = ps(i,j,prev) + Quarter * Omega * ( &
!!$ ps(i-1,j,cur) + ps(i+1,j,prev) + ps(i,j-1,cur) + &
!!$ ps(i,j+1,cur) - Four * ps(i,j,prev) - &
!!$ dx2 * vort(i,j) / mpsi(i,j)**2 )
!!$ end do
!!$ ! (nx-1,ny-1) special
!!$ ps(i,j,cur) = ps(i,j,prev) + Quarter * Omega * ( &
!!$ ps(i-1,j,cur) + ps(i+1,j,cur) + ps(i,j-1,cur) + &
!!$ ps(i,j+1,cur) - Four * ps(i,j,prev) - &
!!$ dx2 * vort(i,j) / mpsi(i,j)**2 )
! Print out psi
!!$ print *, dx
!!$ print *, ps(4:5,10,cur)
!!$ print *, ch(5,10:11,cur)
!!$ print *, vs(5,10)
! Compute boundary values of chi
! do j = 1, ny-1 ! Let ch(nx+1,ny) be fixed at zero
!!!! do j = 1, ny ! Don't fix ch
! ch(nx+1,j,cur) = ch(nx,j,prev) + &
! (ps(nx,j,cur) - ps(nx,j-1,cur)) + dx * us(nx,j) / mu(nx,j)
! end do
!sgd ch(nx+1,1:ny-1,cur) = ch(nx,1:ny-1,prev) + &
!sgd (ps(nx,1:ny-1,cur)-ps(nx,:ny-2,cur)) + dx*us(nx,:ny-1)/mu(nx,:ny-1)
!!$ do i = 1, nx
!!$ ch(i,0,cur) = ch(i,1,prev) + &
!!$ (ps(i,0,cur) - ps(i-1,0,cur)) - dx * vs(i,0) / mv(i,0)
!!$ end do
!sgd ch(1:nx,0,cur) = ch(1:nx,1,prev) + &
!sgd (ps(1:,0,cur) - ps(:nx-1,0,cur)) - dx * vs(:,0) / mv(:,0)
!!$ do j = 1, ny
!!$ ch(0,j,cur) = ch(1,j,prev) - &
!!$ (ps(0,j,cur) - ps(0,j-1,cur)) - dx * us(0,j) / mu(0,j)
!!$ end do
!sgd ch(0,1:ny,cur) = ch(1,1:ny,prev) - &
!sgd (ps(0,1:,cur) - ps(0,:ny-1,cur)) - dx * us(0,:) / mu(0,:)
!!$ do i = 1, nx
!!$ ch(i,ny+1,cur) = ch(i,ny,prev) - &
!!$ (ps(i,ny,cur) - ps(i-1,ny,cur)) + dx * vs(i,ny) / mv(i,ny)
!!$ end do
!sgd ch(1:nx,ny+1,cur) = ch(1:nx,ny,prev) - &
!sgd (ps(1:,ny,cur) - ps(:nx-1,ny,cur)) + dx * vs(:,ny) / mv(:,ny)
! Compute interior values of chi
call update_sor(ch, divForcing, prev, cur)
! Check current errors
! First u
!!$ do j = 1, ny
!!$ do i = 1, nx-1
!!$ u(i,j) = mu(i,j) * (ch(i+1,j,cur) - ch(i,j,cur) - &
!!$ (ps(i,j,cur) - ps(i,j-1,cur))) / dx
!!$ end do
!!$ end do
u = mu(1:nx-1,:) * (ch(2:nx,1:ny,cur) - ch(1:nx-1,1:ny,cur) - &
(ps(1:nx-1,1:,cur) - ps(1:nx-1,:ny-1,cur))) / dx
! Then v
!!$ do j = 1, ny-1
!!$ do i = 1, nx
!!$ v(i,j) = mv(i,j) * (ps(i,j,cur) - ps(i-1,j,cur) + &
!!$ ch(i,j+1,cur) - ch(i,j,cur)) / dx
!!$ end do
!!$ end do
v = mv(:,1:ny-1) * (ps(1:,1:ny-1,cur) - ps(:nx-1,1:ny-1,cur) + &
ch(1:nx,2:ny,cur) - ch(1:nx,1:ny-1,cur)) / dx
err = sqrt(sum(abs(u-us(1:nx-1,:))**2) + sum(abs(v-vs(:,1:ny-1))**2))
! if (mod(k,10) == 0) print *, "Wind:", k, err
if(err < Eps) exit
! Check for errors in vorticity and divergence
!!$ do j = 1, ny-1
!!$ do i = 1, nx-1
!!$ vortCheck(i,j) = mpsi(i,j)**2 * ( &
!!$ ps(i-1,j,cur) + ps(i+1,j,cur) + ps(i,j-1,cur) + &
!!$ ps(i,j+1,cur) - Four * ps(i,j,cur) ) / dx2
!!$ end do
!!$ end do
vortCheck = mpsi**2 * ( &
ps(:nx-2,1:ny-1,cur) + ps(2:,1:ny-1,cur) + ps(1:nx-1,:ny-2,cur) + &
ps(1:nx-1,2:,cur) - Four * ps(1:nx-1,1:ny-1,cur) ) / dx2
! if (mod(k,10) == 0) print *, "Vort:", k, maxval(abs(vortCheck-vort))
!!$ do j = 1, ny
!!$ do i = 1, nx
!!$ divCheck(i,j) = mchi(i,j)**2 * ( &
!!$ ch(i-1,j,cur) + ch(i+1,j,cur) + ch(i,j-1,cur) + &
!!$ ch(i,j+1,cur) - Four * ch(i,j,cur) ) / dx2
!!$ end do
!!$ end do
divCheck = mchi**2 * ( &
ch(:nx-1,1:ny,cur) + ch(2:,1:ny,cur) + ch(1:nx,:ny-1,cur) + &
ch(1:nx,2:,cur) - Four * ch(1:nx,1:ny,cur) ) / dx2
! if (mod(k,10) == 0) print *, "Div: ", k, maxval(abs(divCheck-div))
if (mod(k,250) == 0) write (*, "(a,i6,a,3es12.5)") &
"Iteration", k, ". Errors in Wnd/Vor/Div: ", err, &
maxval(abs(vortCheck-vort)), maxval(abs(divCheck-div))
! Finally, calculate divergence from psi and vorticity from chi
!!$ do j = 1, ny
!!$ do i = 1, nx-1
!!$ u(i,j) = -mu(i,j) * (ps(i,j,cur) - ps(i,j-1,cur)) / dx
!!$ end do
!!$ end do
!!$ do j = 1, ny-1
!!$ do i = 1, nx
!!$ v(i,j) = mv(i,j) * (ps(i,j,cur) - ps(i-1,j,cur)) / dx
!!$ end do
!!$ end do
!!$
!!$ divCheck(1,1) = mchi(1,1)**2 * rdx * ( &
!!$ u(1,1)/mu(1,1) - us(0,1)/mu(0,1) + &
!!$ v(1,1)/mv(1,1) - vs(1,0)/mv(1,0) )
!!$ do i = 2, nx-1
!!$ divCheck(i,1) = mchi(i,1)**2 * rdx * ( &
!!$ u(i,1)/mu(i,1) - u(i-1,1)/mu(i-1,1) + &
!!$ v(i,1)/mv(i,1) - vs(i,0)/mv(i,0) )
!!$ end do
!!$ divCheck(i,1) = mchi(i,1)**2 * rdx * ( &
!!$ us(i,1)/mu(i,1) - u(i-1,1)/mu(i-1,1) + &
!!$ v(i,1)/mv(i,1) - vs(i,0)/mv(i,0) )
!!$ do j = 2, ny-1
!!$ divCheck(1,j) = mchi(1,j)**2 * rdx * ( &
!!$ u(1,j)/mu(1,j) - us(0,j)/mu(0,j) + &
!!$ v(1,j)/mv(1,j) - v(1,j-1)/mv(1,j-1) )
!!$ do i = 2, nx-1
!!$ divCheck(i,j) = mchi(i,j)**2 * rdx * ( &
!!$ u(i,j)/mu(i,j) - u(i-1,j)/mu(i-1,j) + &
!!$ v(i,j)/mv(i,j) - v(i,j-1)/mv(i,j-1) )
!!$ end do
!!$ divCheck(i,j) = mchi(i,j)**2 * rdx * ( &
!!$ us(i,j)/mu(i,j) - u(i-1,j)/mu(i-1,j) + &
!!$ v(i,j)/mv(i,j) - v(i,j-1)/mv(i,j-1) )
!!$ end do
!!$ divCheck(1,j) = mchi(1,j)**2 * rdx * ( &
!!$ u(1,j)/mu(1,j) - us(0,j)/mu(0,j) + &
!!$ vs(1,j)/mv(1,j) - v(1,j-1)/mv(1,j-1) )
!!$ do i = 2, nx-1
!!$ divCheck(i,j) = mchi(i,j)**2 * rdx * ( &
!!$ u(i,j)/mu(i,j) - u(i-1,j)/mu(i-1,j) + &
!!$ vs(i,j)/mv(i,j) - v(i,j-1)/mv(i,j-1) )
!!$ end do
!!$ divCheck(i,j) = mchi(i,j)**2 * rdx * ( &
!!$ us(i,j)/mu(i,j) - u(i-1,j)/mu(i-1,j) + &
!!$ vs(i,j)/mv(i,j) - v(i,j-1)/mv(i,j-1) )
!!$ do j = 1, ny
!!$ do i = 0, nx
!!$ uu(i,j) = -mu(i,j) * (ps(i,j,cur) - ps(i,j-1,cur)) / dx
!!$ end do
!!$ end do
uu = -mu * (ps(:,1:,cur) - ps(:,:ny-1,cur)) / dx
!!$ do j = 0, ny
!!$ do i = 1, nx
!!$ vv(i,j) = mv(i,j) * (ps(i,j,cur) - ps(i-1,j,cur)) / dx
!!$ end do
!!$ end do
vv = mv * (ps(1:,:,cur) - ps(:nx-1,:,cur)) / dx
!!$ do j = 1, ny
!!$ do i = 1, nx
!!$ divCheck(i,j) = mchi(i,j)**2 * rdx * ( &
!!$ uu(i,j)/mu(i,j) - uu(i-1,j)/mu(i-1,j) + &
!!$ vv(i,j)/mv(i,j) - vv(i,j-1)/mv(i,j-1) )
!!$ end do
!!$ end do
divCheck = mchi**2 * rdx * ( &
uu(1:,:)/mu(1:,:) - uu(:nx-1,:)/mu(:nx-1,:) + &
vv(:,1:)/mv(:,1:) - vv(:,:ny-1)/mv(:,:ny-1) )
!!$ do j = 1, ny
!!$ do i = 0, nx
!!$ uu(i,j) = mu(i,j) * (ch(i+1,j,cur) - ch(i,j,cur)) / dx
!!$ end do
!!$ end do
uu = mu * (ch(1:,1:ny,cur) - ch(:nx,1:ny,cur)) / dx
!!$ do j = 0, ny
!!$ do i = 1, nx
!!$ vv(i,j) = mv(i,j) * (ch(i,j+1,cur) - ch(i,j,cur)) / dx
!!$ end do
!!$ end do
vv = mv * (ch(1:nx,1:,cur) - ch(1:nx,:ny,cur)) / dx
!!$ do j = 1, ny-1
!!$ do i = 1, nx-1
!!$ vortCheck(i,j) = mpsi(i,j)**2 * rdx * &
!!$ (vv(i+1,j)/mv(i+1,j) - vv(i,j)/mv(i,j) - &
!!$ (uu(i,j+1)/mu(i,j+1) - uu(i,j)/mu(i,j)))
!!$ end do
!!$ end do
vortCheck = mpsi**2 * rdx * &
(vv(2:,1:ny-1)/mv(2:,1:ny-1)-vv(:nx-1,1:ny-1)/mv(:nx-1,1:ny-1) - &
(uu(1:nx-1,2:)/mu(1:nx-1,2:)-uu(1:nx-1,:ny-1)/mu(1:nx-1,:ny-1)))
if (mod(k,250) == 0) write (*, "(a,2es12.5)") &
" Div from Psi / Vor from Chi: ", maxval(abs(divCheck)), &
maxval(abs(vortCheck))
!!$ print *, vortCheck(1,:)
!!$ print *, vortCheck(5,:)
!!$ print *, nx-1, vortCheck(nx-1,:)
!!$ print *
end do
if (k > MaxIter) stop "Too many iterations!"
!!$! do j = 2, 0, -1
!!$! print *, ps(0:2,j,cur)
!!$! end do
!!$ do j = 2, 1, -1
!!$ print *, uu(0:2,j)
!!$ end do
!!$ do j = 2, 0, -1
!!$ print *, vv(1:2,j)
!!$ end do
!!$! do j = 2, 1, -1
!!$! print *, vort(1:2,j)
!!$! end do
!!$! print *
!!$ do j = 2, 1, -1
!!$ print *, mpsi(1:2,j)
!!$ end do
chi = ch(:,:,cur)
psi = ps(:,:,cur)
end subroutine partition
! ===========================================================================
subroutine update_sor(data, forcing, prev, cur)
real(cp), dimension(0:,0:,:), intent(inout) :: data
real(cp), dimension(:,:), intent(in) :: forcing
integer, intent(in) :: prev, cur
real(cp), parameter :: Omega = 1.8_cp
integer :: nx, ny, j, i
nx = size(forcing,1)
ny = size(forcing,2)
! Sanity checks
if (all(shape(data) /= (/ nx+2, ny+2, 2/))) stop "Mismatch in update_sor!"
if (prev + cur /= 3 .or. cur /= 1 .and. cur /= 2) stop "prev/cur error!"
do j = 1, ny-1
do i = 1, nx-1
data(i,j,cur) = data(i,j,prev) + Quarter * Omega * ( &
data(i-1,j,cur) + data(i+1,j,prev) + data(i,j-1,cur) + &
data(i,j+1,prev) - Four * data(i,j,prev) - forcing(i,j) )
end do
! i = nx special
data(i,j,cur) = data(i,j,prev) + Quarter * Omega * ( &
data(i-1,j,cur) + data(i+1,j,cur) + data(i,j-1,cur) + &
data(i,j+1,prev) - Four * data(i,j,prev) - forcing(i,j) )
end do
! j = ny special
do i = 1, nx-1
data(i,j,cur) = data(i,j,prev) + Quarter * Omega * ( &
data(i-1,j,cur) + data(i+1,j,prev) + data(i,j-1,cur) + &
data(i,j+1,cur) - Four * data(i,j,prev) - forcing(i,j) )
end do
! (nx,ny) special
data(i,j,cur) = data(i,j,prev) + Quarter * Omega * ( &
data(i-1,j,cur) + data(i+1,j,cur) + data(i,j-1,cur) + &
data(i,j+1,cur) - Four * data(i,j,prev) - forcing(i,j) )
end subroutine update_sor
! ===========================================================================
subroutine balance_lhs(psi, f, mCap, m, mu, mv, rdx, lhs)
real(cp), dimension(0:,0:,:), intent(in) :: psi ! (0:nx,0:ny)
real(cp), dimension(:,:), intent(in) :: f, mCap, m ! (1:nx,1:ny)
real(cp), dimension(0:,:), intent(in) :: mu
real(cp), dimension(:,0:), intent(in) :: mv ! (1:nx,0:ny)
real(cp), intent(in) :: rdx
real, dimension(size(f,1),size(f,2),size(psi,3)), intent(out) :: lhs
character(len=*), parameter :: Mismatch = "mismatch in balance_lhs!"
real(cp), dimension(:,:), allocatable :: mdxp, mdyp, c, fmvdxp, fmudyp, t1, &
mdx1, mdx2, t2, mdy1, mdy2, t3
integer :: nx, ny, nz, k, i, j
nx = size(psi,1) - 1
ny = size(psi,2) - 1
nz = size(psi,3)
! Sanity checks
if (any(shape(f) /= (/ nx, ny /) )) stop "f " // Mismatch
if (any(shape(m) /= (/ nx, ny /) )) stop "m " // Mismatch
if (any(shape(mu) /= (/ nx+1, ny /) )) stop "mu " // Mismatch
if (any(shape(mv) /= (/ nx, ny+1 /) )) stop "mv " // Mismatch
allocate(mdxp(nx,0:ny), mdyp(0:nx,ny), c(nx-1,ny-1))
allocate(fmvdxp(nx-1,2:ny-1), fmudyp(2:nx-1,ny-1))
allocate(t1(2:nx-1,2:ny-1), t2(2:nx-1,2:ny-1), t3(2:nx-1,2:ny-1))
allocate(mdx1(nx-1,2:ny-1), mdx2(nx-1,2:ny-1))
allocate(mdy1(2:nx-1,ny-1), mdy2(2:nx-1,ny-1))
lhs = Zero
do k = 1, nz
mdxp = mv * xdiff(psi(:,:,k), rdx)
mdyp = mu * ydiff(psi(:,:,k), rdx)
fmvdxp = xavg(f(:,2:ny-1)) * xyavg(mdxp(:,1:ny-1)) / mu(1:nx-1,2:ny-1)
fmudyp = yavg(f(2:nx-1,:)) * xyavg(mdyp(1:nx-1,:)) / mv(2:nx-1,1:ny-1)
t1 = m(2:nx-1,2:ny-1)**2 * (xdiff(fmvdxp, rdx) + ydiff(fmudyp, rdx))
!!$ mdx1 = mu(1:nx-1,2:ny-1) * mdyp(1:nx-1,2:ny-1) * xdiff(xavg(mdyp(:,2:ny-1)) / mCap(:,2:ny-1), rdx)
forall (i = 1:nx-1, j = 2:ny-1)
mdx1(i,j) = Half * mu(i,j)**2 * rdx**3 &
* ( mu(i,j) * (One/mCap(i+1,j) - One/mCap(i,j)) * psi(i,j,k)**2 &
- Two * mu(i,j) * (One/mCap(i+1,j) - One/mCap(i,j)) * psi(i,j-1,k) &
* psi(i,j,k) &
+ mu(i,j) * (One/mCap(i+1,j) - One/mCap(i,j)) * psi(i,j-1,k)**2 &
+ mu(i+1,j)/mCap(i+1,j) * psi(i,j,k) * psi(i+1,j,k) &
- mu(i+1,j)/mCap(i+1,j) * psi(i,j-1,k) * psi(i+1,j,k) &
- mu(i+1,j)/mCap(i+1,j) * psi(i,j,k) * psi(i+1,j-1,k) &
+ mu(i+1,j)/mCap(i+1,j) * psi(i,j-1,k) * psi(i+1,j-1,k) &
- mu(i-1,j)/mCap(i,j) * psi(i,j,k) * psi(i-1,j,k) &
+ mu(i-1,j)/mCap(i,j) * psi(i,j-1,k) * psi(i-1,j,k) &
+ mu(i-1,j)/mCap(i,j) * psi(i,j,k) * psi(i-1,j-1,k) &
- mu(i-1,j)/mCap(i,j) * psi(i,j-1,k) * psi(i-1,j-1,k) )
end forall
!!$ mdx2 = mu(1:nx-1,2:ny-1) * xyavg(mdxp(:,1:ny-1)) * ydiff(yavg(mdyp(1:nx-1,:)) &
!!$ / xyavg(mCap), rdx)
forall (i = 1:nx-1, j = 1:ny-1)
c(i,j) = One / (mCap(i,j) + mCap(i+1,j) + mCap(i,j+1) + mCap(i+1,j+1))
end forall
forall (i = 1:nx-1, j = 2:ny-1)
mdx2(i,j) = Half * mu(i,j) * rdx**3 &
* ( mv(i,j-1) * psi(i,j-1,k) - mv(i,j-1) * psi(i-1,j-1,k) &
+ mv(i+1,j-1) * psi(i+1,j-1,k) - mv(i+1,j-1) * psi(i,j-1,k) &
+ mv(i,j) * psi(i,j,k) - mv(i,j) * psi(i-1,j,k) &
+ mv(i+1,j) * psi(i+1,j,k) - mv(i+1,j) * psi(i,j,k) ) &
* ( c(i,j) * mu(i,j+1) * psi(i,j+1,k) - c(i,j) * mu(i,j+1) * psi(i,j,k) &
+ c(i,j) * mu(i,j) * psi(i,j,k) - c(i,j) * mu(i,j) * psi(i,j-1,k) &
- c(i,j-1) * mu(i,j) * psi(i,j,k) + c(i,j-1) * mu(i,j) * psi(i,j-1,k) &
- c(i,j-1) * mu(i,j-1) * psi(i,j-1,k) &
+ c(i,j-1) * mu(i,j-1) * psi(i,j-2,k) )
end forall
! t2 = m(2:nx-1,2:ny-1) * xdiff(mdx1 - mdx2, rdx)
forall (i = 2:nx-1, j = 2:ny-1)
t2(i,j) = Half * rdx**4 * m(i,j) * ( &
mu(i,j) * (mu(i,j) * (psi(i,j,k) - psi(i,j-1,k)) * ( &
mu(i,j) * (One/mCap(i+1,j)-One/mCap(i,j)) * (psi(i,j,k) - psi(i,j-1,k)) +&
mu(i+1,j)/mCap(i+1,j) * (psi(i+1,j,k) - psi(i+1,j-1,k)) - &
mu(i-1,j)/mCap(i,j) * (psi(i-1,j,k) - psi(i-1,j-1,k))) - ( &
mv(i,j-1) * (psi(i,j-1,k) - psi(i-1,j-1,k)) + &
mv(i+1,j-1) * (psi(i+1,j-1,k) - psi(i,j-1,k)) + &
mv(i,j) * (psi(i,j,k) - psi(i-1,j,k)) + &
mv(i+1,j) * (psi(i+1,j,k) - psi(i,j,k))) * ( &
c(i,j) * (mu(i,j+1) * (psi(i,j+1,k) - psi(i,j,k)) + &
mu(i,j) * (psi(i,j,k) - psi(i,j-1,k))) - &
c(i,j-1) * (mu(i,j) * (psi(i,j,k) - psi(i,j-1,k)) + &
mu(i,j-1) * (psi(i,j-1,k) - psi(i,j-2,k))))) - &
mu(i-1,j) * (mu(i-1,j) * (psi(i-1,j,k) - psi(i-1,j-1,k)) * ( &
mu(i-1,j)*(One/mCap(i,j)-One/mCap(i-1,j))*(psi(i-1,j,k)-psi(i-1,j-1,k)) +&
mu(i,j)/mCap(i,j) * (psi(i,j,k) - psi(i,j-1,k)) - &
mu(i-2,j)/mCap(i-1,j) * (psi(i-2,j,k) - psi(i-2,j-1,k))) - ( &
mv(i-1,j-1) * (psi(i-1,j-1,k) - psi(i-2,j-1,k)) + &
mv(i,j-1) * (psi(i,j-1,k) - psi(i-1,j-1,k)) + &
mv(i-1,j) * (psi(i-1,j,k) - psi(i-2,j,k)) + &
mv(i,j) * (psi(i,j,k) - psi(i-1,j,k))) * ( &
c(i-1,j) * (mu(i-1,j+1) * (psi(i-1,j+1,k) - psi(i-1,j,k)) + &
mu(i-1,j) * (psi(i-1,j,k) - psi(i-1,j-1,k))) - &
c(i-1,j-1) * (mu(i-1,j) * (psi(i-1,j,k) - psi(i-1,j-1,k)) + &
mu(i-1,j-1) * (psi(i-1,j-1,k) - psi(i-1,j-2,k))))))
end forall
mdy1 = mv(2:nx-1,1:ny-1) * mdxp(2:nx-1,1:ny-1) * ydiff(yavg(mdxp(2:nx-1,:)) &
/ mCap(2:nx-1,:), rdx)
mdy2 = mv(2:nx-1,1:ny-1) * xyavg(mdyp(1:nx-1,:)) * xdiff(xavg(mdxp(:,1:ny-1)) &
/ xyavg(mCap), rdx)
t3 = m(2:nx-1,2:ny-1) * ydiff(mdy1 - mdy2, rdx)
lhs(2:nx-1,2:ny-1,k) = t1 - t2 - t3
end do
deallocate(mdy2, mdy1, mdx2, mdx1, t3, t2, t1, fmudyp, fmvdxp, mdyp, mdxp)
end subroutine balance_lhs
! ===========================================================================
subroutine balance_lhs_exp(psi, f, mCap, m, mu, mv, rdx, lhs)
real(cp), dimension(0:,0:,:), intent(in) :: psi ! (0:nx,0:ny)
real(cp), dimension(:,:), intent(in) :: f, mCap, m ! (1:nx,1:ny)
real(cp), dimension(0:,:), intent(in) :: mu
real(cp), dimension(:,0:), intent(in) :: mv ! (1:nx,0:ny)
real(cp), intent(in) :: rdx
real, dimension(size(f,1),size(f,2),size(psi,3)), intent(out) :: lhs
character(len=*), parameter :: Mismatch = "mismatch in balance_lhs!"
real(cp), dimension(:,:,:), allocatable :: t1, t2, t3
real(cp), dimension(:,:), allocatable :: c
integer :: nx, ny, nz, i, j, k
nx = size(psi,1) - 1
ny = size(psi,2) - 1
nz = size(psi,3)
! Sanity checks
if (any(shape(f) /= (/ nx, ny /) )) stop "f " // Mismatch
if (any(shape(m) /= (/ nx, ny /) )) stop "m " // Mismatch
if (any(shape(mu) /= (/ nx+1, ny /) )) stop "mu " // Mismatch
if (any(shape(mv) /= (/ nx, ny+1 /) )) stop "mv " // Mismatch
allocate(t1(2:nx-1,2:ny-1,nz), t2(2:nx-1,2:ny-1,nz), t3(2:nx-1,2:ny-1,nz))
allocate(c(nx-1,ny-1))
forall (i = 1:nx-1, j = 1:ny-1)
c(i,j) = One / (mCap(i,j) + mCap(i+1,j) + mCap(i,j+1) + mCap(i+1,j+1))
end forall
lhs = Zero
forall (i = 2:nx-1, j = 2:ny-1, k = 1:nz)
t1(i,j,k) = Eighth * rdx**2 * m(i,j)**2 * ( &
(f(i, j) + f(i+1,j)) * ( &
mv(i, j-1) * (psi(i, j-1,k) - psi(i-1,j-1,k)) + &
mv(i+1,j-1) * (psi(i+1,j-1,k) - psi(i, j-1,k)) + &
mv(i, j ) * (psi(i, j, k) - psi(i-1,j, k)) + &
mv(i+1,j ) * (psi(i+1,j, k) - psi(i, j, k)) ) / mu(i, j) - &
(f(i-1,j) + f(i,j )) * ( &
mv(i-1,j-1) * (psi(i-1,j-1,k) - psi(i-2,j-1,k)) + &
mv(i, j-1) * (psi(i, j-1,k) - psi(i-1,j-1,k)) + &
mv(i-1,j ) * (psi(i-1,j, k) - psi(i-2,j, k)) + &
mv(i, j ) * (psi(i, j, k) - psi(i-1,j, k)) ) / mu(i-1,j) + &
(f(i, j) + f(i,j+1)) * ( &
mu(i-1,j ) * (psi(i-1,j, k) - psi(i-1,j-1,k)) + &
mu(i, j ) * (psi(i, j, k) - psi(i, j-1,k)) + &
mu(i-1,j+1) * (psi(i-1,j+1,k) - psi(i-1,j, k)) + &
mu(i, j+1) * (psi(i, j+1,k) - psi(i, j, k)) ) / mv(i, j) - &
(f(i,j-1) + f(i,j )) * ( &
mu(i-1,j-1) * (psi(i-1,j-1,k) - psi(i-1,j-2,k)) + &
mu(i, j-1) * (psi(i, j-1,k) - psi(i, j-2,k)) + &
mu(i-1,j ) * (psi(i-1,j, k) - psi(i-1,j-1,k)) + &
mu(i, j ) * (psi(i, j, k) - psi(i, j-1,k)) ) / mv(i,j-1) )
t2(i,j,k) = Half * rdx**4 * m(i,j) * ( &
mu(i,j) * (mu(i,j) * (psi(i,j,k) - psi(i,j-1,k)) &
* (mu(i,j) * (One/mCap(i+1,j)-One/mCap(i,j)) * (psi(i,j,k) - psi(i,j-1,k)) &
+ mu(i+1,j)/mCap(i+1,j) * (psi(i+1,j,k) - psi(i+1,j-1,k)) &
- mu(i-1,j)/mCap(i,j) * (psi(i-1,j,k) - psi(i-1,j-1,k))) &
- (mv(i,j-1) * (psi(i,j-1,k) - psi(i-1,j-1,k)) &
+ mv(i+1,j-1) * (psi(i+1,j-1,k) - psi(i,j-1,k)) &
+ mv(i,j) * (psi(i,j,k) - psi(i-1,j,k)) + mv(i+1,j) * (psi(i+1,j,k) - psi(i,j,k))) &
* (c(i,j) * (mu(i,j+1) * (psi(i,j+1,k) - psi(i,j,k)) + mu(i,j) &
* (psi(i,j,k) - psi(i,j-1,k))) - c(i,j-1) * (mu(i,j) * (psi(i,j,k) - psi(i,j-1,k)) &
+ mu(i,j-1) * (psi(i,j-1,k) - psi(i,j-2,k))))) &
- mu(i-1,j) * (mu(i-1,j) * (psi(i-1,j,k) - psi(i-1,j-1,k)) &
* (mu(i-1,j) * (One/mCap(i,j)-One/mCap(i-1,j)) * (psi(i-1,j,k) - psi(i-1,j-1,k)) &
+ mu(i,j)/mCap(i,j) * (psi(i,j,k) - psi(i,j-1,k)) &
- mu(i-2,j)/mCap(i-1,j) * (psi(i-2,j,k) - psi(i-2,j-1,k))) &
- (mv(i-1,j-1) * (psi(i-1,j-1,k) - psi(i-2,j-1,k)) &
+ mv(i,j-1) * (psi(i,j-1,k) - psi(i-1,j-1,k)) &
+ mv(i-1,j) * (psi(i-1,j,k) - psi(i-2,j,k)) + mv(i,j) * (psi(i,j,k) - psi(i-1,j,k))) &
* (c(i-1,j) * (mu(i-1,j+1) * (psi(i-1,j+1,k) - psi(i-1,j,k)) &
+ mu(i-1,j) * (psi(i-1,j,k) - psi(i-1,j-1,k))) &
- c(i-1,j-1) * (mu(i-1,j) * (psi(i-1,j,k) - psi(i-1,j-1,k)) &
+ mu(i-1,j-1) * (psi(i-1,j-1,k) - psi(i-1,j-2,k))))))
t3(i,j,k) = Half * rdx**4 * m(i,j) * ( &
mv(i,j) * (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) * (One/mCap(i,j+1) &
* (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) + mv(i,j+1) * (psi(i,j+1,k)-psi(i-1,j+1,k))) &
- One/mCap(i,j) * (mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)) &
+ mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)))) - (mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k)) &
+ mu(i,j) * (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j+1) * (psi(i-1,j+1,k)-psi(i-1,j,k)) &
+ mu(i,j+1) * (psi(i,j+1,k)-psi(i,j,k))) * (c(i,j) &
* (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) + mv(i+1,j) * (psi(i+1,j,k)-psi(i,j,k))) &
- c(i-1,j) * (mv(i-1,j) * (psi(i-1,j,k)-psi(i-2,j,k)) &
+ mv(i,j) * (psi(i,j,k)-psi(i-1,j,k))))) &
- mv(i,j-1) * (mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)) * (One/mCap(i,j) &
* (mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)) + mv(i,j) * (psi(i,j,k)-psi(i-1,j,k))) &
- One/mCap(i,j-1) * (mv(i,j-2) * (psi(i,j-2,k)-psi(i-1,j-2,k)) &
+ mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)))) &
- (mu(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-1,j-2,k)) &
+ mu(i,j-1) * (psi(i,j-1,k)-psi(i,j-2,k)) + mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k)) &
+ mu(i,j) * (psi(i,j,k)-psi(i,j-1,k))) * (c(i,j-1) * (mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k)) + mv(i+1,j-1) * (psi(i+1,j-1,k)-psi(i,j-1,k))) &
- c(i-1,j-1) * (mv(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-2,j-1,k)) &
+ mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))))))
end forall
lhs(2:nx-1,2:ny-1,:) = t1 - t2 - t3
deallocate(c, t3, t2, t1)
end subroutine balance_lhs_exp
! ===========================================================================
subroutine balance_lhs_tlm(dps, psi, f, mCap, m, mu, mv, rdx, dlhs)
real(cp), dimension(0:,0:,:), intent(in) :: dps, psi ! (0:nx,0:ny)
real(cp), dimension(:,:), intent(in) :: f, mCap, m ! (1:nx,1:ny)
real(cp), dimension(0:,:), intent(in) :: mu
real(cp), dimension(:,0:), intent(in) :: mv ! (1:nx,0:ny)
real(cp), intent(in) :: rdx
real, dimension(size(f,1),size(f,2),size(dps,3)), intent(out) :: dlhs
character(len=*), parameter :: Mismatch = "mismatch in balance_lhs_tlm!"
real(cp), dimension(:,:,:), allocatable :: dpsi, t1, t2, t3
real(cp), dimension(:,:), allocatable :: c
integer :: nx, ny, nz, i, j, k
nx = size(dps,1) - 1
ny = size(dps,2) - 1
nz = size(dps,3)
! Sanity checks
if (any(shape(f) /= (/ nx, ny /) )) stop "f " // Mismatch
if (any(shape(m) /= (/ nx, ny /) )) stop "m " // Mismatch
if (any(shape(mu) /= (/ nx+1, ny /) )) stop "mu " // Mismatch
if (any(shape(mv) /= (/ nx, ny+1 /) )) stop "mv " // Mismatch
allocate(t1(2:nx-1,2:ny-1,nz), t2(2:nx-1,2:ny-1,nz), t3(2:nx-1,2:ny-1,nz))
allocate(dpsi(0:nx,0:ny,nz))
allocate(c(nx-1,ny-1))
dpsi = dps
dpsi((/0,nx/),:,:) = Zero
dpsi(1:nx-1,(/0,ny/),:) = Zero
dlhs = Zero ! Really just setting the i/j boundaries.
forall (i = 1:nx-1, j = 1:ny-1)
c(i,j) = One / (mCap(i,j) + mCap(i+1,j) + mCap(i,j+1) + mCap(i+1,j+1))
end forall
forall (i = 2:nx-1, j = 2:ny-1, k = 1:nz)
t1(i,j,k) = m(i,j)**2 * Eighth * rdx**2 * ( &
dpsi(i-1,j-2,k) * mu(i-1,j-1) / mv(i,j-1) * (f(i,j-1) + f(i,j)) + &
dpsi(i, j-2,k) * mu(i, j-1) / mv(i,j-1) * (f(i,j-1) + f(i,j)) + &
dpsi(i-2,j-1,k) * mv(i-1,j-1) / mu(i-1,j) * (f(i-1,j) + f(i,j)) - &
dpsi(i-1,j-1,k) * ( mv(i,j-1) / mu(i, j) * (f(i,j) + f(i+1,j)) + &
(f(i-1,j) + f(i,j)) / mu(i-1,j) * (mv(i-1,j-1) - mv(i,j-1)) + &
mu(i-1,j) / mv(i, j) * (f(i,j) + f(i,j+1)) + &
(f(i,j-1) + f(i,j)) / mv(i,j-1) * (mu(i-1,j-1) - mu(i-1,j))) + &
dpsi(i, j-1,k) * (-mv(i,j-1) / mu(i-1,j) * (f(i-1,j) + f(i,j)) + &
(f(i,j) + f(i+1,j)) / mu(i, j) * (mv(i,j-1) - mv(i+1,j-1)) - &
mu(i, j) / mv(i, j) * (f(i,j) + f(i,j+1)) + &
(f(i,j-1) + f(i,j)) / mv(i,j-1) * (mu(i, j) - mu(i, j-1))) + &
dpsi(i+1,j-1,k) * mv(i+1,j-1) / mu(i, j) * (f(i,j) + f(i+1,j)) + &
dpsi(i-2,j, k) * mv(i-1, j) / mu(i-1,j) * (f(i-1,j) + f(i,j)) - &
dpsi(i-1,j, k) * ( mv(i, j) / mu(i, j) * (f(i,j) + f(i+1,j)) + &
(f(i-1,j) + f(i,j)) / mu(i-1,j) * (mv(i-1,j) - mv(i, j)) + &
mu(i-1,j) / mv(i,j-1) * (f(i,j-1) + f(i,j)) + &
(f(i,j) + f(i,j+1)) / mv(i, j) * (mu(i-1,j+1) - mu(i-1,j))) + &
dpsi(i, j, k) * (-mv(i, j) / mu(i-1,j) * (f(i-1,j) + f(i,j)) + &
(f(i,j) + f(i+1,j)) / mu(i, j) * (mv(i, j) - mv(i+1,j)) - &
mu(i, j) / mv(i,j-1) * (f(i,j-1) + f(i,j)) + &
(f(i,j) + f(i,j+1)) / mv(i, j) * (mu(i, j) - mu(i,j+1))) + &
dpsi(i+1,j, k) * mv(i+1, j) / mu(i, j) * (f(i,j) + f(i+1,j)) + &
dpsi(i-1,j+1,k) * mu(i-1,j+1) / mv(i, j) * (f(i,j) + f(i,j+1)) + &
dpsi(i, j+1,k) * mu(i, j+1) / mv(i, j) * (f(i,j) + f(i,j+1)))
t2(i,j,k) = m(i,j) * Half * rdx**4 * ( &
dpsi(i-1,j-2,k) * c(i-1,j-1) * mu(i-1,j-1) * mu(i-1,j) * ( &
(psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) + (psi(i-1,j,k)-psi(i-2,j,k)) * mv(i-1,j) + &
(psi(i,j-1,k)-psi(i-1,j-1,k))*mv(i,j-1) + (psi(i-1,j-1,k)-psi(i-2,j-1,k))*mv(i-1,j-1)))
t2(i,j,k) = t2(i,j,k) - m(i,j) * Half * rdx**4 * ( &
dpsi(i,j-2,k) * c(i,j-1) * mu(i,j-1) * mu(i,j) * ( &
(psi(i+1,j,k)-psi(i,j,k)) * mv(i+1,j) + (psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) + &
(psi(i+1,j-1,k)-psi(i,j-1,k))*mv(i+1,j-1) + (psi(i,j-1,k)-psi(i-1,j-1,k))*mv(i,j-1)))
t2(i,j,k) = t2(i,j,k) - m(i,j) * Half * rdx**4 * ( &
dpsi(i-2,j-1,k) * mu(i-1,j) * ((c(i-1,j) * ( &
(psi(i-1,j+1,k)-psi(i-1,j,k))*mu(i-1,j+1) + (psi(i-1,j,k)-psi(i-1,j-1,k))*mu(i-1,j)) &
- c(i-1,j-1) * ((psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j) + &
(psi(i-1,j-1,k)-psi(i-1,j-2,k))*mu(i-1,j-1))) * mv(i-1,j-1) &
+ mu(i-2,j)*mu(i-1,j)/mCap(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k))))
t2(i,j,k) = t2(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i-1,j-1,k) * (mu(i,j) * (( &
c(i,j) * ((psi(i,j+1,k)-psi(i,j,k))*mu(i,j+1) + (psi(i,j,k)-psi(i,j-1,k))*mu(i,j)) - &
c(i,j-1) * ((psi(i,j,k)-psi(i,j-1,k))*mu(i,j) + (psi(i,j-1,k)-psi(i,j-2,k))*mu(i,j-1) &
)) * mv(i,j-1) + mu(i-1,j)*mu(i,j)/mCap(i,j) * (psi(i,j,k)-psi(i,j-1,k))) &
- mu(i-1,j) * ((c(i-1,j) * mu(i-1,j) + c(i-1,j-1) * (mu(i-1,j-1) - mu(i-1,j))) * ( &
(psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) + (psi(i-1,j,k)-psi(i-2,j,k)) * mv(i-1,j) + &
(psi(i,j-1,k)-psi(i-1,j-1,k))*mv(i,j-1) + (psi(i-1,j-1,k)-psi(i-2,j-1,k))*mv(i-1,j-1))&
- (c(i-1,j) * ((psi(i-1,j+1,k)-psi(i-1,j,k)) * mu(i-1,j+1) &
+ (psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j)) &
- c(i-1,j-1) * ((psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j) &
+ (psi(i-1,j-1,k)-psi(i-1,j-2,k)) * mu(i-1,j-1))) &
* (mv(i-1,j-1) - mv(i,j-1)) - mu(i-1,j) * (mu(i,j)/mCap(i,j) &
* (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (One/mCap(i,j) - One/mCap(i-1,j)) &
* (psi(i-1,j,k)-psi(i-1,j-1,k))-mu(i-2,j)/mCap(i-1,j)*(psi(i-2,j,k)-psi(i-2,j-1,k))) &
- mu(i-1,j)**2 * (One/mCap(i,j) - One/mCap(i-1,j)) * (psi(i-1,j,k)-psi(i-1,j-1,k)))))
t2(i,j,k) = t2(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i,j-1,k) * (mu(i,j) * ((c(i,j) * mu(i,j) + c(i,j-1) * (mu(i,j-1) - mu(i,j))) * &
((psi(i+1,j,k)-psi(i,j,k)) * mv(i+1,j) + (psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) + &
(psi(i+1,j-1,k)-psi(i,j-1,k))*mv(i+1,j-1) + (psi(i,j-1,k)-psi(i-1,j-1,k))*mv(i,j-1)) &
- (c(i,j) * ((psi(i,j+1,k)-psi(i,j,k))*mu(i,j+1) + (psi(i,j,k)-psi(i,j-1,k))*mu(i,j)) &
- c(i,j-1)*((psi(i,j,k)-psi(i,j-1,k))*mu(i,j)+(psi(i,j-1,k)-psi(i,j-2,k))*mu(i,j-1))) &
* (mv(i,j-1) - mv(i+1,j-1)) &
- mu(i,j) * (mu(i+1,j)/mCap(i+1,j) * (psi(i+1,j,k)-psi(i+1,j-1,k)) &
+ mu(i,j) * (One/mCap(i+1,j) - One/mCap(i,j)) * (psi(i,j,k)-psi(i,j-1,k)) &
- mu(i-1,j)/mCap(i,j) * (psi(i-1,j,k)-psi(i-1,j-1,k))) - mu(i,j)**2 &
* (One/mCap(i+1,j) - One/mCap(i,j)) * (psi(i,j,k)-psi(i,j-1,k))) - mu(i-1,j) * ( -( &
c(i-1,j) * ((psi(i-1,j+1,k)-psi(i-1,j,k)) * mu(i-1,j+1) &
+ (psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j)) - c(i-1,j-1) &
* ((psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j) &
+ (psi(i-1,j-1,k)-psi(i-1,j-2,k)) * mu(i-1,j-1))) * mv(i,j-1) &
- mu(i-1,j)*mu(i,j)/mCap(i,j) * (psi(i-1,j,k)-psi(i-1,j-1,k)))))
t2(i,j,k) = t2(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i+1,j-1,k) * mu(i,j) * ( -(c(i,j) * ((psi(i,j+1,k)-psi(i,j,k)) * mu(i,j+1) &
+ (psi(i,j,k)-psi(i,j-1,k))*mu(i,j)) - c(i,j-1) * ((psi(i,j,k)-psi(i,j-1,k))*mu(i,j) &
+ (psi(i,j-1,k)-psi(i,j-2,k)) * mu(i,j-1))) * mv(i+1,j-1) &
- mu(i,j)*mu(i+1,j)/mCap(i+1,j) * (psi(i,j,k)-psi(i,j-1,k))))
t2(i,j,k) = t2(i,j,k) - m(i,j) * Half * rdx**4 * ( &
dpsi(i-2,j,k) * mu(i-1,j) * ((c(i-1,j) * ((psi(i-1,j+1,k)-psi(i-1,j,k)) * mu(i-1,j+1) &
+ (psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j)) - c(i-1,j-1) &
* ((psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j) &
+ (psi(i-1,j-1,k)-psi(i-1,j-2,k)) * mu(i-1,j-1))) * mv(i-1,j) &
- mu(i-2,j)*mu(i-1,j)/mCap(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k))))
t2(i,j,k) = t2(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i-1,j,k) * (mu(i,j) * ((c(i,j) * ((psi(i,j+1,k)-psi(i,j,k)) * mu(i,j+1) &
+ (psi(i,j,k)-psi(i,j-1,k))*mu(i,j)) - c(i,j-1) * ((psi(i,j,k)-psi(i,j-1,k))*mu(i,j) &
+ (psi(i,j-1,k)-psi(i,j-2,k)) * mu(i,j-1))) * mv(i,j) - mu(i-1,j)*mu(i,j)/mCap(i,j) &
* (psi(i,j,k)-psi(i,j-1,k))) - mu(i-1,j) * (-(c(i-1,j) * (mu(i-1,j) - mu(i-1,j+1)) &
- c(i-1,j-1) * mu(i-1,j)) * ((psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) &
+ (psi(i-1,j,k)-psi(i-2,j,k)) * mv(i-1,j) + (psi(i,j-1,k)-psi(i-1,j-1,k)) * mv(i,j-1) &
+ (psi(i-1,j-1,k)-psi(i-2,j-1,k)) * mv(i-1,j-1)) &
- (c(i-1,j) * ((psi(i-1,j+1,k)-psi(i-1,j,k)) * mu(i-1,j+1) &
+ (psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j)) &
- c(i-1,j-1) * ((psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j) &
+ (psi(i-1,j-1,k)-psi(i-1,j-2,k)) * mu(i-1,j-1))) * (mv(i-1,j) - mv(i,j)) &
+ mu(i-1,j) * (mu(i,j)/mCap(i,j) * (psi(i,j,k)-psi(i,j-1,k)) &
+ mu(i-1,j) * (One/mCap(i,j) - One/mCap(i-1,j)) * (psi(i-1,j,k)-psi(i-1,j-1,k)) &
- mu(i-2,j)/mCap(i-1,j) * (psi(i-2,j,k)-psi(i-2,j-1,k))) &
+ mu(i-1,j)**2 * (One/mCap(i,j) - One/mCap(i-1,j)) * (psi(i-1,j,k)-psi(i-1,j-1,k)))))
t2(i,j,k) = t2(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i,j,k) * (mu(i,j) * (-(c(i,j) * (mu(i,j)-mu(i,j+1)) - c(i,j-1)*mu(i,j)) &
* ((psi(i+1,j,k)-psi(i,j,k)) * mv(i+1,j) + (psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) &
+ (psi(i+1,j-1,k)-psi(i,j-1,k))*mv(i+1,j-1) + (psi(i,j-1,k)-psi(i-1,j-1,k))*mv(i,j-1))&
- (c(i,j) * ((psi(i,j+1,k)-psi(i,j,k))*mu(i,j+1) + (psi(i,j,k)-psi(i,j-1,k))*mu(i,j)) &
- c(i,j-1) * ((psi(i,j,k)-psi(i,j-1,k)) * mu(i,j) &
+ (psi(i,j-1,k)-psi(i,j-2,k)) * mu(i,j-1))) * (mv(i,j) - mv(i+1,j)) + mu(i,j) &
* (mu(i+1,j)/mCap(i+1,j) * (psi(i+1,j,k)-psi(i+1,j-1,k)) + mu(i,j) &
* (One/mCap(i+1,j) - One/mCap(i,j)) * (psi(i,j,k)-psi(i,j-1,k)) - mu(i-1,j)/mCap(i,j) &
* (psi(i-1,j,k)-psi(i-1,j-1,k))) + mu(i,j)**2 * (One/mCap(i+1,j) - One/mCap(i,j)) &
* (psi(i,j,k)-psi(i,j-1,k))) - mu(i-1,j) * (mu(i-1,j)*mu(i,j)/mCap(i,j) &
* (psi(i-1,j,k)-psi(i-1,j-1,k)) - (c(i,j) * ((psi(i-1,j+1,k)-psi(i-1,j,k))*mu(i-1,j+1)&
+ (psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j)) - c(i-1,j-1) &
* ((psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j) &
+ (psi(i-1,j-1,k)-psi(i-1,j-2,k)) * mu(i-1,j-1))) * mv(i,j))))
t2(i,j,k) = t2(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i+1,j,k) * mu(i,j) * (mu(i,j)*mu(i+1,j)/mCap(i+1,j) * (psi(i,j,k)-psi(i,j-1,k)) &
- (c(i,j) * ((psi(i,j+1,k)-psi(i,j,k))*mu(i,j+1) + (psi(i,j,k)-psi(i,j-1,k))*mu(i,j)) &
- c(i,j-1) * ((psi(i,j,k)-psi(i,j-1,k)) * mu(i,j) &
+ (psi(i,j-1,k)-psi(i,j-2,k)) * mu(i,j-1))) * mv(i+1,j)))
t2(i,j,k) = t2(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i-1,j+1,k) * c(i-1,j) * mu(i-1,j) * mu(i-1,j+1) &
* ((psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) + (psi(i-1,j,k)-psi(i-2,j,k)) * mv(i-1,j) &
+ (psi(i,j-1,k)-psi(i-1,j-1,k))*mv(i,j-1)+(psi(i-1,j-1,k)-psi(i-2,j-1,k))*mv(i-1,j-1)))
t2(i,j,k) = t2(i,j,k) - m(i,j) * Half * rdx**4 * ( &
dpsi(i,j+1,k) * c(i,j) * mu(i,j) * mu(i,j+1) * ((psi(i+1,j,k)-psi(i,j,k)) * mv(i+1,j) &
+ (psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) + (psi(i+1,j-1,k)-psi(i,j-1,k)) * mv(i+1,j-1) &
+ (psi(i,j-1,k)-psi(i-1,j-1,k)) * mv(i,j-1)))
t3(i,j,k) = m(i,j) * Half * rdx**4 * ( &
-dpsi(i-1,j-2,k) * mv(i,j-1) * (mu(i-1,j-1) * (c(i,j-1) * (mv(i+1,j-1) &
* (psi(i+1,j-1,k)-psi(i,j-1,k)) + mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) &
- c(i-1,j-1) * (mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)) &
+ mv(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-2,j-1,k)))) &
+ mv(i,j-2)/mCap(i,j-1) * mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))))
t3(i,j,k) = t3(i,j,k) - m(i,j) * Half * rdx**4 * ( &
dpsi(i,j-2,k) * mv(i,j-1) * (mu(i,j-1) * (c(i,j-1) * (mv(i+1,j-1) &
* (psi(i+1,j-1,k)-psi(i,j-1,k)) + mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) &
- c(i-1,j-1) * (mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)) &
+ mv(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-2,j-1,k)))) &
- mv(i,j-2)/mCap(i,j-1) * mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))))
t3(i,j,k) = t3(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i-2,j-1,k) * c(i-1,j-1) * mv(i-1,j-1) * mv(i,j-1) * (mu(i,j) &
* (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k)) + mu(i,j-1) &
* (psi(i,j-1,k)-psi(i,j-2,k)) + mu(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-1,j-2,k))))
t3(i,j,k) = t3(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i-1,j-1,k) * (mv(i,j) * (mu(i-1,j) &
* (c(i,j) * (mv(i+1,j)*(psi(i+1,j,k)-psi(i,j,k)) + mv(i,j)*(psi(i,j,k)-psi(i-1,j,k))) &
- c(i-1,j)*(mv(i,j)*(psi(i,j,k)-psi(i-1,j,k))+mv(i-1,j)*(psi(i-1,j,k)-psi(i-2,j,k)))) &
+ mv(i,j)*mv(i,j-1)/mCap(i,j) * (psi(i,j,k)-psi(i-1,j,k))) - mv(i,j-1) * (-mv(i,j-1) &
* (One/mCap(i,j) * (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) + mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k))) - One/mCap(i,j-1) * (mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k)) + mv(i,j-2) * (psi(i,j-2,k)-psi(i-1,j-2,k)))) &
- (mu(i-1,j-1) - mu(i-1,j)) * (c(i,j-1) * (mv(i+1,j-1) * (psi(i+1,j-1,k)-psi(i,j-1,k))&
+ mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) - c(i-1,j-1) * (mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k)) + mv(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-2,j-1,k)))) &
+ mv(i,j-1)**2 * (psi(i,j-1,k)-psi(i-1,j-1,k)) * (One/mCap(i,j-1) - One/mCap(i,j)) &
+ (mu(i,j) * (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k)) &
+ mu(i,j-1)*(psi(i,j-1,k)-psi(i,j-2,k)) + mu(i-1,j-1)*(psi(i-1,j-1,k)-psi(i-1,j-2,k)))&
* (c(i,j-1) * mv(i,j-1) + c(i-1,j-1) * (mv(i-1,j-1) - mv(i,j-1))))))
t3(i,j,k) = t3(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i,j-1,k) * (mv(i,j) * (mu(i,j) &
* (c(i,j) * (mv(i+1,j)*(psi(i+1,j,k)-psi(i,j,k)) + mv(i,j)*(psi(i,j,k)-psi(i-1,j,k))) &
- c(i-1,j)*(mv(i,j)*(psi(i,j,k)-psi(i-1,j,k))+mv(i-1,j)*(psi(i-1,j,k)-psi(i-2,j,k)))) &
- mv(i,j)*mv(i,j-1)/mCap(i,j) * (psi(i,j,k)-psi(i-1,j,k))) - mv(i,j-1) * (mv(i,j-1) &
* (One/mCap(i,j) * (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) + mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k))) - One/mCap(i,j-1) * (mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k)) + mv(i,j-2) * (psi(i,j-2,k)-psi(i-1,j-2,k)))) &
- (mu(i,j-1) - mu(i,j)) * (c(i,j-1) * (mv(i+1,j-1) * (psi(i+1,j-1,k)-psi(i,j-1,k)) &
+ mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) - c(i-1,j-1) * (mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k)) + mv(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-2,j-1,k)))) &
- (mu(i,j) * (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k)) &
+ mu(i,j-1)*(psi(i,j-1,k)-psi(i,j-2,k)) + mu(i-1,j-1)*(psi(i-1,j-1,k)-psi(i-1,j-2,k)))&
* (c(i,j-1) * (mv(i,j-1) - mv(i+1,j-1)) - c(i-1,j-1) * mv(i,j-1)) &
+ m(i,j-1)**2 * (psi(i,j-1,k)-psi(i-1,j-1,k)) * (One/mCap(i,j) - One/mCap(i,j-1)))))
t3(i,j,k) = t3(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i+1,j-1,k) * c(i,j-1) * mv(i,j-1) * mv(i+1,j-1) * (mu(i,j) &
* (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k)) + mu(i,j-1) &
* (psi(i,j-1,k)-psi(i,j-2,k)) + mu(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-1,j-2,k))))
t3(i,j,k) = t3(i,j,k) - m(i,j) * Half * rdx**4 * ( &
dpsi(i-2,j,k) * c(i-1,j) * mv(i-1,j) * mv(i,j) &
* (mu(i,j+1) * (psi(i,j+1,k)-psi(i,j,k)) + mu(i-1,j+1) * (psi(i-1,j+1,k)-psi(i-1,j,k))&
+ mu(i,j) * (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k))))
t3(i,j,k) = t3(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i-1,j,k) * (mv(i,j) * (-(mu(i-1,j) - mu(i-1,j+1)) * (c(i,j) * (mv(i+1,j) &
* (psi(i+1,j,k)-psi(i,j,k)) + mv(i,j) * (psi(i,j,k)-psi(i-1,j,k))) - c(i-1,j) &
* (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) + mv(i-1,j) * (psi(i-1,j,k)-psi(i-2,j,k)))) &
- mv(i,j) * (One/mCap(i,j+1) * (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) &
+ mv(i,j+1) * (psi(i,j+1,k)-psi(i-1,j+1,k))) - One/mCap(i,j) &
* (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) + mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)))) &
+ mv(i,j)**2 * (psi(i,j,k)-psi(i-1,j,k)) * (One/mCap(i,j) - One/mCap(i,j+1)) &
- (mu(i,j+1) * (psi(i,j+1,k)-psi(i,j,k)) + mu(i-1,j+1) * (psi(i-1,j+1,k)-psi(i-1,j,k))&
+ mu(i,j) * (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k))) &
* (-c(i,j) * mv(i,j) - c(i-1,j) * (mv(i-1,j) - mv(i,j)))) &
- mv(i,j-1) * (-mv(i,j-1)*mv(i,j)/mCap(i,j) * (psi(i,j-1,k)-psi(i-1,j-1,k)) &
- mu(i-1,j) * (c(i,j-1) * (mv(i+1,j-1) * (psi(i+1,j-1,k)-psi(i,j-1,k)) &
+ mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) - c(i-1,j-1) * (mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k)) + mv(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-2,j-1,k)))))))
t3(i,j,k) = t3(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i,j,k) * (mv(i,j) * (-(mu(i,j) - mu(i,j+1)) * (c(i,j) * (mv(i+1,j) &
* (psi(i+1,j,k)-psi(i,j,k)) + mv(i,j) * (psi(i,j,k)-psi(i-1,j,k))) - c(i-1,j) &
* (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) + mv(i-1,j) * (psi(i-1,j,k)-psi(i-2,j,k)))) &
- (mu(i,j+1) * (psi(i,j+1,k)-psi(i,j,k)) + mu(i-1,j+1) * (psi(i-1,j+1,k)-psi(i-1,j,k))&
+ mu(i,j) * (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k))) &
* (c(i,j) * (mv(i,j) - mv(i+1,j)) - c(i-1,j) * mv(i,j)) + mv(i,j) * (One/mCap(i,j+1) &
* (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) + mv(i,j+1) * (psi(i,j+1,k)-psi(i-1,j+1,k))) &
- One/mCap(i,j) * (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) &
+ mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)))) - mv(i,j)**2 * (psi(i,j,k)-psi(i-1,j,k))&
* (One/mCap(i,j) - One/mCap(i,j+1))) - mv(i,j-1) * (mv(i,j-1)*mv(i,j)/mCap(i,j) &
* (psi(i,j-1,k)-psi(i-1,j-1,k)) - mu(i,j) * (c(i,j-1) * (mv(i+1,j-1) &
* (psi(i+1,j-1,k)-psi(i,j-1,k)) + mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) &
- c(i-1,j-1) * (mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)) &
+ mv(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-2,j-1,k)))))))
t3(i,j,k) = t3(i,j,k) - m(i,j) * Half * rdx**4 * ( &
dpsi(i+1,j,k) * c(i,j) * mv(i,j) * mv(i+1,j) &
* (mu(i,j+1) * (psi(i,j+1,k)-psi(i,j,k)) + mu(i-1,j+1) * (psi(i-1,j+1,k)-psi(i-1,j,k))&
+ mu(i,j) * (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k))))
t3(i,j,k) = t3(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i-1,j+1,k) * mv(i,j) * (-mu(i-1,j+1) * (c(i,j) &
* (mv(i+1,j) * (psi(i+1,j,k)-psi(i,j,k)) + mv(i,j) * (psi(i,j,k)-psi(i-1,j,k))) &
- c(i-1,j) * (mv(i,j) * (psi(i,j,k)-psi(i-1,j,k)) &
+ mv(i-1,j) * (psi(i-1,j,k)-psi(i-2,j,k)))) &
- mv(i,j)*mv(i,j+1)/mCap(i,j+1) * (psi(i,j,k)-psi(i-1,j,k))))
t3(i,j,k) = t3(i,j,k) + m(i,j) * Half * rdx**4 * ( &
dpsi(i,j+1,k) * mv(i,j) * (mv(i,j)*mv(i,j+1)/mCap(i,j+1) * (psi(i,j,k)-psi(i-1,j,k)) &
- mu(i,j+1) * (c(i,j) * (mv(i+1,j) * (psi(i+1,j,k)-psi(i,j,k)) &
+ mv(i,j)*(psi(i,j,k)-psi(i-1,j,k))) - c(i-1,j) * (mv(i,j)*(psi(i,j,k)-psi(i-1,j,k)) &
+ mv(i-1,j) * (psi(i-1,j,k)-psi(i-2,j,k))))))
end forall
dlhs(2:nx-1,2:ny-1,:) = t1 - t2 - t3
deallocate(c, dpsi, t3, t2, t1)
end subroutine balance_lhs_tlm
! ===========================================================================
subroutine balance_lhs_adj(t1, psi, f, mCap, m, mu, mv, rdx, ps)
real(cp), dimension(:,:,:), intent(in) :: t1 ! (1:nx,1:ny)
real(cp), dimension(0:,0:,:), intent(in) :: psi ! (0:nx,0:ny)
real(cp), dimension(:,:), intent(in) :: f, mCap, m ! (1:nx,1:ny)
real(cp), dimension(0:,:), intent(in) :: mu
real(cp), dimension(:,0:), intent(in) :: mv ! (1:nx,0:ny)
real(cp), intent(in) :: rdx
real(cp), dimension(0:size(t1,1),0:size(t1,2),size(t1,3)), intent(out) :: ps
character(len=*), parameter :: Mismatch = "mismatch in balance_lhs_adj!"
real(cp), dimension(:,:), allocatable :: c, cc, dd
integer :: nx, ny, nz, i, j, k
nx = size(t1,1)
ny = size(t1,2)
nz = size(t1,3)
! Sanity checks
if (any(shape(psi) /= (/ nx+1, ny+1, nz /) )) stop "psi " // Mismatch
if (any(shape(f) /= (/ nx, ny /) )) stop "f " // Mismatch
if (any(shape(mCap) /= (/ nx, ny /) )) stop "mCap " // Mismatch
if (any(shape(m) /= (/ nx, ny /) )) stop "m " // Mismatch
if (any(shape(mu) /= (/ nx+1, ny /) )) stop "mu " // Mismatch
if (any(shape(mv) /= (/ nx, ny+1 /) )) stop "mv " // Mismatch
allocate(c(nx-1,ny-1), cc(nx,ny), dd(nx,ny))
forall (i = 1:nx-1, j = 1:ny-1)
c(i,j) = One / (mCap(i,j) + mCap(i+1,j) + mCap(i,j+1) + mCap(i+1,j+1))
end forall
ps = Zero
cc = m**2 * Eighth * rdx**2
dd = m * Half * rdx**4
forall (i = 2:nx-1, j = 2:ny-1, k = 1:nz)
ps(i-1,j-2,k) = ps(i-1,j-2,k) + cc(i,j) * &
mu(i-1,j-1) / mv(i,j-1) * (f(i,j-1) + f(i, j)) * t1(i,j,k)
ps(i-1,j-2,k) = ps(i-1,j-2,k) - dd(i,j) * &
c(i-1,j-1) * mu(i-1,j-1) * mu(i-1,j) * ((psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) &
+ (psi(i-1,j,k)-psi(i-2,j,k)) * mv(i-1,j) + (psi(i,j-1,k)-psi(i-1,j-1,k)) * mv(i,j-1) &
+ (psi(i-1,j-1,k)-psi(i-2,j-1,k)) * mv(i-1,j-1)) * t1(i,j,k)
ps(i-1,j-2,k) = ps(i-1,j-2,k) + dd(i,j) * &
mv(i,j-1) * (mu(i-1,j-1) * (c(i,j-1) * (mv(i+1,j-1) * (psi(i+1,j-1,k)-psi(i,j-1,k)) &
+ mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) - c(i-1,j-1) * (mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k)) + mv(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-2,j-1,k)))) &
+ mv(i,j-2)/mCap(i,j-1) * mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) * t1(i,j,k)
ps(i ,j-2,k) = ps(i ,j-2,k) + cc(i,j) * &
mu(i, j-1) / mv(i,j-1) * (f(i,j-1) + f(i, j)) * t1(i,j,k)
ps(i ,j-2,k) = ps(i ,j-2,k) + dd(i,j) * &
c(i,j-1) * mu(i,j-1) * mu(i,j) * ((psi(i+1,j,k)-psi(i,j,k)) * mv(i+1,j) &
+ (psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) + (psi(i+1,j-1,k)-psi(i,j-1,k)) * mv(i+1,j-1) &
+ (psi(i,j-1,k)-psi(i-1,j-1,k)) * mv(i,j-1)) * t1(i,j,k)
ps(i ,j-2,k) = ps(i ,j-2,k) + dd(i,j) * &
mv(i,j-1) * (mu(i,j-1) * (c(i,j-1) * (mv(i+1,j-1) * (psi(i+1,j-1,k)-psi(i,j-1,k)) &
+ mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) - c(i-1,j-1) * (mv(i,j-1) &
* (psi(i,j-1,k)-psi(i-1,j-1,k)) + mv(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-2,j-1,k)))) &
- mv(i,j-2)/mCap(i,j-1) * mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) * t1(i,j,k)
ps(i-2,j-1,k) = ps(i-2,j-1,k) + cc(i,j) * &
mv(i-1,j-1) / mu(i-1,j) * (f(i-1,j) + f(i, j)) * t1(i,j,k)
ps(i-2,j-1,k) = ps(i-2,j-1,k) + dd(i,j) * &
mu(i-1,j) * ((c(i-1,j) * ((psi(i-1,j+1,k)-psi(i-1,j,k)) * mu(i-1,j+1) &
+ (psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j)) - c(i-1,j-1) &
* ((psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j) &
+ (psi(i-1,j-1,k)-psi(i-1,j-2,k)) * mu(i-1,j-1))) * mv(i-1,j-1) &
+ mu(i-2,j)*mu(i-1,j)/mCap(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k))) * t1(i,j,k)
ps(i-2,j-1,k) = ps(i-2,j-1,k) - dd(i,j) * &
c(i-1,j-1) * mv(i-1,j-1) * mv(i,j-1) * (mu(i,j) * (psi(i,j,k)-psi(i,j-1,k)) &
+ mu(i-1,j) * (psi(i-1,j,k)-psi(i-1,j-1,k)) + mu(i,j-1) * (psi(i,j-1,k)-psi(i,j-2,k)) &
+ mu(i-1,j-1) * (psi(i-1,j-1,k)-psi(i-1,j-2,k))) * t1(i,j,k)
ps(i-1,j-1,k) = ps(i-1,j-1,k) - cc(i,j) * ( &
mv(i, j-1) / mu(i, j) * (f(i,j) + f(i+1,j)) + &
(f(i-1,j) + f(i,j)) / mu(i-1,j) * (mv(i-1,j-1) - mv(i,j-1)) + &
mu(i-1, j) / mv(i, j) * (f(i,j) + f(i,j+1)) + &
(f(i,j-1) + f(i,j)) / mv(i,j-1) * (mu(i-1,j-1) - mu(i-1,j)) ) &
* t1(i,j,k)
ps(i-1,j-1,k) = ps(i-1,j-1,k) - dd(i,j) * &
(mu(i,j) * ((c(i,j) * ((psi(i,j+1,k)-psi(i,j,k)) * mu(i,j+1) &
+ (psi(i,j,k)-psi(i,j-1,k)) * mu(i,j)) - c(i,j-1) * ((psi(i,j,k)-psi(i,j-1,k))*mu(i,j)&
+ (psi(i,j-1,k)-psi(i,j-2,k)) * mu(i,j-1))) * mv(i,j-1) + mu(i-1,j)*mu(i,j)/mCap(i,j) &
* (psi(i,j,k)-psi(i,j-1,k))) - mu(i-1,j) * ((c(i-1,j) * mu(i-1,j) + c(i-1,j-1) &
* (mu(i-1,j-1) - mu(i-1,j))) * ((psi(i,j,k)-psi(i-1,j,k)) * mv(i,j) &
+ (psi(i-1,j,k)-psi(i-2,j,k)) * mv(i-1,j) + (psi(i,j-1,k)-psi(i-1,j-1,k)) * mv(i,j-1) &
+ (psi(i-1,j-1,k)-psi(i-2,j-1,k)) * mv(i-1,j-1)) - (c(i-1,j) &
* ((psi(i-1,j+1,k)-psi(i-1,j,k)) * mu(i-1,j+1) &
+ (psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j)) &
- c(i-1,j-1) * ((psi(i-1,j,k)-psi(i-1,j-1,k)) * mu(i-1,j) &
+ (psi(i-1,j-1,k)-psi(i-1,j-2,k)) * mu(i-1,j-1))) &
* (mv(i-1,j-1) - mv(i,j-1)) - mu(i-1,j) * (mu(i,j)/mCap(i,j) &
* (psi(i,j,k)-psi(i,j-1,k)) + mu(i-1,j) * (One/mCap(i,j) - One/mCap(i-1,j)) &
* (psi(i-1,j,k)-psi(i-1,j-1,k))-mu(i-2,j)/mCap(i-1,j)*(psi(i-2,j,k)-psi(i-2,j-1,k))) &
- mu(i-1,j)**2 * (One/mCap(i,j) - One/mCap(i-1,j)) * (psi(i-1,j,k)-psi(i-1,j-1,k)))) &
* t1(i,j,k)
ps(i-1,j-1,k) = ps(i-1,j-1,k) - dd(i,j) * &
(mv(i,j) * (mu(i-1,j) * (c(i,j) * (mv(i+1,j)*(psi(i+1,j,k)-psi(i,j,k)) &
+ mv(i,j)*(psi(i,j,k)-psi(i-1,j,k))) - c(i-1,j)*(mv(i,j)*(psi(i,j,k)-psi(i-1,j,k)) &
+ mv(i-1,j)*(psi(i-1,j,k)-psi(i-2,j,k)))) + mv(i,j)*mv(i,j-1)/mCap(i,j) &
* (psi(i,j,k)-psi(i-1,j,k))) - mv(i,j-1) * (-mv(i,j-1) * (One/mCap(i,j) * (mv(i,j) &
* (psi(i,j,k)-psi(i-1,j,k)) + mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k))) &
- One/mCap(i,j-1) * (mv(i,j-1) * (psi(i,j-1,k)-psi(i-1,j-1,k)) &
+ mv(i,j-2) * (psi(i,j-2,k)-psi(i-1,j-2,k)))) - (mu(i-1,j-1) - mu(i-1,j)) * (c(i,j-1) &