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finite_differences.f90
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finite_differences.f90
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module finite_differences
implicit none
public :: first_order_upwind
public :: third_order_upwind
public :: fifth_order_upwind
public :: third_order_upwind_zed
public :: first_order_upwind_zed
public :: second_order_centered
public :: fourth_order_centered
public :: second_order_centered_zed
public :: four_point_triangle
public :: fd3pt, fd5pt
public :: d2_3pt
public :: fd_variable_upwinding_vpa
public :: fd_variable_upwinding_zed
public :: fd_cell_centres_zed, cell_centres_zed
interface fd3pt
module procedure fd3pt_real
module procedure fd3pt_real_array
module procedure fd3pt_complex_array
end interface
interface fd5pt
module procedure fd5pt_real
module procedure fd5pt_array
end interface
interface first_order_upwind
module procedure first_order_upwind_real
module procedure first_order_upwind_complex
end interface
interface third_order_upwind
module procedure third_order_upwind_complex
module procedure third_order_upwind_real
end interface
interface fifth_order_upwind
module procedure fifth_order_upwind_complex
module procedure fifth_order_upwind_real
end interface
interface tridag
module procedure tridag_real
module procedure tridag_complex
end interface
interface second_order_centered
module procedure second_order_centered_real
module procedure second_order_centered_complex
end interface
interface four_point_triangle
module procedure four_point_triangle_real
module procedure four_point_triangle_complex
end interface
interface fourth_order_centered
module procedure fourth_order_centered_real
module procedure fourth_order_centered_complex
end interface
interface second_order_centered_zed
module procedure second_order_centered_zed_real
module procedure second_order_centered_zed_complex
end interface
interface d2_3pt
module procedure d2_3pt_real
module procedure d2_3pt_complex
end interface
contains
subroutine first_order_upwind_real (llim, f, del, sgn, df)
implicit none
integer, intent (in) :: llim
real, dimension (llim:), intent (in) :: f
real, intent (in) :: del
integer, intent (in) :: sgn
real, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
if (sgn == -1) then
istart = llim
iend = llim+n-1
else
istart = llim+n-1
iend = llim
end if
! zero BC, 1st order accurate upwind
df(istart) = -f(istart)*sgn/del
do i = istart-sgn, iend, -sgn
df(i) = sgn*(f(i+sgn)-f(i))/del
end do
end subroutine first_order_upwind_real
subroutine first_order_upwind_complex (llim, f, del, sgn, df)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
integer, intent (in) :: sgn
complex, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
if (sgn == -1) then
istart = llim
iend = llim+n-1
else
istart = llim+n-1
iend = llim
end if
! zero BC, 1st order accurate upwind
df(istart) = -f(istart)*sgn/del
do i = istart-sgn, iend, -sgn
df(i) = sgn*(f(i+sgn)-f(i))/del
end do
end subroutine first_order_upwind_complex
subroutine third_order_upwind_complex (llim, f, del, sgn, df)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
integer, intent (in) :: sgn
complex, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
if (sgn == -1) then
istart = llim
iend = llim+n-1
else
istart = llim+n-1
iend = llim
end if
i = istart-sgn
! zero BC, 1st order accurate upwind
df(istart) = -f(istart)*sgn/del
! zero BC, 3rd order accurate upwind
df(i) = -sgn*(2.*f(i-sgn)+3.*f(i)-6.*f(i+sgn))/(6.*del)
! 1st order accurate upwind
df(iend) = sgn*(f(iend+sgn)-f(iend))/del
! 3rd order accurate upwind
do i = istart-2*sgn, iend+sgn, -sgn
df(i) = -sgn*(2.*f(i-sgn)+3*f(i)-6.*f(i+sgn)+f(i+2*sgn))/(6.*del)
end do
end subroutine third_order_upwind_complex
subroutine third_order_upwind_real (llim, f, del, sgn, df)
implicit none
integer, intent (in) :: llim
real, dimension (llim:), intent (in) :: f
real, intent (in) :: del
integer, intent (in) :: sgn
real, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
if (sgn == -1) then
istart = llim
iend = llim+n-1
else
istart = llim+n-1
iend = llim
end if
i = istart-sgn
! zero BC, 1st order accurate upwind
df(istart) = -f(istart)*sgn/del
! zero BC, 3rd order accurate upwind
df(i) = -sgn*(2.*f(i-sgn)+3.*f(i)-6.*f(i+sgn))/(6.*del)
! 1st order accurate upwind
df(iend) = sgn*(f(iend+sgn)-f(iend))/del
! 3rd order accurate upwind
do i = istart-2*sgn, iend+sgn, -sgn
df(i) = -sgn*(2.*f(i-sgn)+3*f(i)-6.*f(i+sgn)+f(i+2*sgn))/(6.*del)
end do
end subroutine third_order_upwind_real
subroutine fifth_order_upwind_complex (llim, f, del, sgn, df)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
integer, intent (in) :: sgn
complex, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
if (sgn == -1) then
istart = llim
iend = llim+n-1
else
istart = llim+n-1
iend = llim
end if
! zero BC, 1st order accurate upwind
df(istart) = -f(istart)*sgn/del
! zero BC, 3rd order accurate upwind
i = istart-sgn
df(i) = -sgn*(2.*f(i-sgn)+3.*f(i)-6.*f(i+sgn))/(6.*del)
! zero BC, 5th order accurate upwind
i = istart-2*sgn
df(i) = -sgn*(-3.*f(i-2*sgn)+30.*f(i-sgn)+20.*f(i)-60.*f(i+sgn)+15.*f(i+2*sgn))/(60.*del)
! 1st order accurate upwind
df(iend) = sgn*(f(iend+sgn)-f(iend))/del
! 3rd order accurate upwind
df(iend+sgn) = -sgn*(2.*f(iend)+3*f(iend+sgn)-6.*f(iend+2*sgn)+f(iend+3*sgn))/(6.*del)
! 5th order accurate upwind
do i = istart-3*sgn, iend+2*sgn, -sgn
df(i) = -sgn*(-3.*f(i-2*sgn)+30.*f(i-sgn)+20.*f(i)-60.*f(i+sgn)+15.*f(i+2*sgn)-2.*f(i+3*sgn)) &
/(60.*del)
end do
end subroutine fifth_order_upwind_complex
subroutine fifth_order_upwind_real (llim, f, del, sgn, df)
implicit none
integer, intent (in) :: llim
real, dimension (llim:), intent (in) :: f
real, intent (in) :: del
integer, intent (in) :: sgn
real, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
if (sgn == -1) then
istart = llim
iend = llim+n-1
else
istart = llim+n-1
iend = llim
end if
! zero BC, 1st order accurate upwind
df(istart) = -f(istart)*sgn/del
! zero BC, 3rd order accurate upwind
i = istart-sgn
df(i) = -sgn*(2.*f(i-sgn)+3.*f(i)-6.*f(i+sgn))/(6.*del)
! zero BC, 5th order accurate upwind
i = istart-2*sgn
df(i) = -sgn*(-3.*f(i-2*sgn)+30.*f(i-sgn)+20.*f(i)-60.*f(i+sgn)+15.*f(i+2*sgn))/(60.*del)
! 1st order accurate upwind
df(iend) = -sgn*(f(iend)-f(iend+sgn))/del
! 3rd order accurate upwind
df(iend+sgn) = -sgn*(2.*f(iend)+3*f(iend+sgn)-6.*f(iend+2*sgn)+f(iend+3*sgn))/(6.*del)
! 5th order accurate upwind
do i = istart-3*sgn, iend+2*sgn, -sgn
df(i) = -sgn*(-3.*f(i-2*sgn)+30.*f(i-sgn)+20.*f(i)-60.*f(i+sgn)+15.*f(i+2*sgn)-2.*f(i+3*sgn)) &
/(60.*del)
end do
end subroutine fifth_order_upwind_real
subroutine third_order_upwind_zed (llim, iseg, nseg, f, del, sgn, fl, fr, periodic, df)
implicit none
integer, intent (in) :: llim, iseg, nseg
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
integer, intent (in) :: sgn
logical, intent (in) :: periodic
complex, dimension (:), intent (in) :: fl, fr
complex, dimension (llim:), intent (out) :: df
integer :: i, istart, iend, ulim
ulim = size(f)+llim-1
! if sgn > 0, then stream speed is negative
! so sweep from more positive to more negative zed
if (sgn > 0) then
if (iseg == nseg .and..not.periodic) then
i = ulim
df(i) = -f(i)/del
i = ulim-1
df(i) = -(2.*f(i-1)+3.*f(i)-6.*f(i+1))/(6.*del)
else
i = ulim
df(i) = -(2.*f(i-1)+3.*f(i)-6.*fr(1)+fr(2))/(6.*del)
i = ulim-1
df(i) = -(2.*f(i-1)+3.*f(i)-6.*f(i+1)+fr(1))/(6.*del)
end if
if (iseg == 1.and..not.periodic) then
i = llim
df(i) = (f(i+1)-f(i))/del
else
i = llim
df(i) = -(2.*fl(2)+3*f(i)-6.*f(i+1)+f(i+2))/(6.*del)
end if
istart = ulim
iend = llim
else
if (iseg == 1.and..not.periodic) then
i = llim
df(i) = f(i)/del
i = llim+1
df(i) = (2.*f(i+1)+3.*f(i)-6.*f(i-1))/(6.*del)
else
i = llim
df(i) = (2.*f(i+1)+3*f(i)-6.*fl(2)+fl(1))/(6.*del)
i = llim+1
df(i) = (2.*f(i+1)+3*f(i)-6.*f(i-1)+fl(2))/(6.*del)
end if
if (iseg == nseg.and..not.periodic) then
i = ulim
df(i) = (f(i)-f(i-1))/del
else
i = ulim
df(i) = (2.*fr(1)+3*f(i)-6.*f(i-1)+f(i-2))/(6.*del)
end if
istart = llim
iend = ulim
end if
! 3rd order accurate upwind
do i = istart-2*sgn, iend+sgn, -sgn
df(i) = -sgn*(2.*f(i-sgn)+3*f(i)-6.*f(i+sgn)+f(i+2*sgn))/(6.*del)
end do
end subroutine third_order_upwind_zed
subroutine first_order_upwind_zed (llim, iseg, nseg, f, del, sgn, fl, fr, periodic, df)
implicit none
integer, intent (in) :: llim, iseg, nseg
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
integer, intent (in) :: sgn
logical, intent (in) :: periodic
complex, dimension (:), intent (in) :: fl, fr
complex, dimension (llim:), intent (out) :: df
integer :: i, istart, iend, ulim
ulim = size(f)+llim-1
! if sgn > 0, then stream speed is negative
! so sweep from more positive to more negative zed
if (sgn > 0) then
if (iseg == nseg.and..not.periodic) then
i = ulim
df(i) = -f(i)/del
i = ulim-1
df(i) = (f(i+1)-f(i))/del
else
i = ulim
df(i) = (fr(1)-f(i))/del
i = ulim-1
df(i) = (f(i+1)-f(i))/del
end if
i = llim
df(i) = (f(i+1)-f(i))/del
istart = ulim
iend = llim
else
if (iseg == 1.and..not.periodic) then
i = llim
df(i) = f(i)/del
i = llim+1
df(i) = (f(i)-f(i-1))/del
else
i = llim
df(i) = (f(i)-fl(2))/del
i = llim+1
df(i) = (f(i)-f(i-1))/del
end if
i = ulim
df(i) = (f(i)-f(i-1))/del
istart = llim
iend = ulim
end if
! 3rd order accurate upwind
do i = istart-2*sgn, iend+sgn, -sgn
df(i) = sgn*(f(i+sgn)-f(i))/del
end do
end subroutine first_order_upwind_zed
subroutine second_order_centered_real (llim, f, del, df)
implicit none
integer, intent (in) :: llim
real, dimension (llim:), intent (in) :: f
real, intent (in) :: del
real, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
istart = llim
iend = llim+n-1
! zero BC
df(istart) = f(istart+1) / (2.*del)
df(iend) =-f(iend-1) / (2.*del)
! 2nd order accurate centered
do i = istart+1, iend-1
df(i) = (f(i+1) - f(i-1)) / (2.*del)
end do
end subroutine second_order_centered_real
subroutine second_order_centered_complex (llim, f, del, df)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
complex, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
istart = llim
iend = llim+n-1
! zero BC
df(istart) = f(istart+1) / (2.*del)
df(iend) =-f(iend-1) / (2.*del)
! 2nd order accurate centered
do i = istart+1, iend-1
df(i) = (f(i+1) - f(i-1)) / (2.*del)
end do
end subroutine second_order_centered_complex
subroutine four_point_triangle_real (llim, f, del, df)
implicit none
integer, intent (in) :: llim
real, dimension (llim:), intent (in) :: f
real, intent (in) :: del
real, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
istart = llim
iend = llim+n-1
! zero BC
i=istart
df(i) = f(i+1) / (2.0*del)
i=istart+1
df(i) = (f(i+1)-f(i-1)) / (2.0*del)
i=istart+2
df(i) = (-2.*f(i+3)+9.*f(i+1)-9.*f(i-1)) / (18.0*del)
i=iend
df(i) = -f(i-1) / (2.0*del)
i=iend-1
df(i) = (f(i+1)-f(i-1)) / (2.0*del)
i=iend-2
df(i) = (9.*f(i+1)-9.*f(i-1)+2.*f(i-3)) / (18.0*del)
! 2nd order accurate centered
do i = istart+3, iend-3
df(i) = (-2.*f(i+3)+9.*f(i+1)-9.*f(i-1)+2.*f(i-3)) / (18.0*del)
end do
end subroutine four_point_triangle_real
subroutine four_point_triangle_complex (llim, f, del, df)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
complex, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
istart = llim
iend = llim+n-1
! zero BC
i=istart
df(i) = f(i+1) / (2.0*del)
i=istart+1
df(i) = (f(i+1)-f(i-1)) / (2.0*del)
i=istart+2
df(i) = (-2.*f(i+3)+9.*f(i+1)-9.*f(i-1)) / (18.0*del)
i=iend
df(i) = -f(i-1) / (2.0*del)
i=iend-1
df(i) = (f(i+1)-f(i-1)) / (2.0*del)
i=iend-2
df(i) = (9.*f(i+1)-9.*f(i-1)+2.*f(i-3)) / (18.0*del)
! 2nd order accurate centered
do i = istart+3, iend-3
df(i) = (-2.*f(i+3)+9.*f(i+1)-9.*f(i-1)+2.*f(i-3)) / (18.0*del)
end do
end subroutine four_point_triangle_complex
subroutine fourth_order_centered_real (llim, f, del, df)
implicit none
integer, intent (in) :: llim
real, dimension (llim:), intent (in) :: f
real, intent (in) :: del
real, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
istart = llim
iend = llim+n-1
! zero BC
! 2nd order accurate centered
df(istart) = f(istart + 1) / (2.*del)
df(iend) =-f(iend - 1) / (2.*del)
! 4th order accurate centered
df(istart+1) = (f(istart+3) - 8.*f(istart+2) + 8.*f(istart)) / (12.*del)
df(iend-1) = (-8*f(iend) + 8.*f(iend-2) - f(iend-3)) / (12.*del)
! 4th order accurate centered
do i = istart+2, iend-2
df(i) = (f(i+2)-8.*f(i+1) + 8.*f(i-1)-f(i-2)) / (12.*del)
end do
end subroutine fourth_order_centered_real
subroutine fourth_order_centered_complex (llim, f, del, df)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
complex, dimension (llim:), intent (out) :: df
integer :: i, n, istart, iend
n = size(f)
istart = llim
iend = llim+n-1
! zero BC
! 2nd order accurate centered
df(istart) = f(istart + 1) / (2.*del)
df(iend) =-f(iend - 1) / (2.*del)
! 4th order accurate centered
df(istart+1) = (f(istart+3) - 8.*f(istart+2) + 8.*f(istart))/ (12.*del)
df(iend-1) = (-8*f(iend) + 8.*f(iend-2) - f(iend-3)) / (12.*del)
! 4th order accurate centered
do i = istart+2, iend-2
df(i) = (f(i+2)-8.*f(i+1) + 8.*f(i-1)-f(i-2)) / (12.*del)
end do
end subroutine fourth_order_centered_complex
subroutine second_order_centered_zed_real (llim, iseg, nseg, f, del, sgn, fl, fr, periodic, df)
implicit none
integer, intent (in) :: llim, iseg, nseg
real, dimension (llim:), intent (in) :: f
integer, intent (in) :: sgn
real, intent (in) :: del
real, dimension (:), intent (in) :: fl, fr
logical, intent (in) :: periodic
real, dimension (llim:), intent (out) :: df
integer :: i, ulim
ulim = size(f)+llim-1
i = llim
if (iseg == 1 .and. sgn>0 .and..not.periodic) then
! sgn > 0 corresponds to negative advection speed
! upwind at boundary requires taking information from right
df(i) = (f(i+1)-f(i))/del
else
df(i) = 0.5*(f(i+1)-fl(2))/del
end if
i = ulim
if (iseg == nseg .and. sgn<0 .and..not.periodic) then
! sgn < 0 corresponds to positive advection speed
! upwind at boundary requires taking information from left
df(i) = (f(i)-f(i-1))/del
else
df(i) = 0.5*(fr(1)-f(i-1))/del
end if
do i = llim+1, ulim-1
df(i) = 0.5*(f(i+1)-f(i-1))/del
end do
end subroutine second_order_centered_zed_real
subroutine second_order_centered_zed_complex (llim, iseg, nseg, f, del, sgn, fl, fr, periodic, df)
implicit none
integer, intent (in) :: llim, iseg, nseg
complex, dimension (llim:), intent (in) :: f
integer, intent (in) :: sgn
real, intent (in) :: del
complex, dimension (:), intent (in) :: fl, fr
logical, intent (in) :: periodic
complex, dimension (llim:), intent (out) :: df
integer :: i, ulim
ulim = size(f)+llim-1
i = llim
if (iseg == 1 .and. sgn>0 .and..not.periodic) then
! sgn > 0 corresponds to negative advection speed
! upwind at boundary requires taking information from right
df(i) = (f(i+1)-f(i))/del
else
df(i) = 0.5*(f(i+1)-fl(2))/del
end if
i = ulim
if (iseg == nseg .and. sgn<0 .and..not.periodic) then
! sgn < 0 corresponds to positive advection speed
! upwind at boundary requires taking information from left
df(i) = (f(i)-f(i-1))/del
else
df(i) = 0.5*(fr(1)-f(i-1))/del
end if
do i = llim+1, ulim-1
df(i) = 0.5*(f(i+1)-f(i-1))/del
end do
end subroutine second_order_centered_zed_complex
subroutine second_order_centered_vpa (llim, f, del, df)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
complex, dimension (llim:), intent (out) :: df
integer :: i, ulim
ulim = size(f)+llim-1
i = llim
df(i) = 0.5*f(i+1)/del
i = ulim
df(i) = -0.5*f(i-1)/del
do i = llim+1, ulim-1
df(i) = 0.5*(f(i+1)-f(i-1))/del
end do
end subroutine second_order_centered_vpa
subroutine fd_cell_centres_zed (llim, f, del, sgn, fl, fr, df)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del
integer, intent (in) :: sgn
complex, intent (in) :: fl, fr
complex, dimension (llim:), intent (out) :: df
integer :: i, ulim
ulim = size(f)+llim-1
if (sgn > 0) then
! if sgn > 0, then stream speed is negative
! so sweep from more positive to more negative zed
i = ulim
df(i) = (fr-f(i))/del
do i = ulim-1, llim, -1
df(i) = (f(i+1)-f(i))/del
end do
else
! if sgn < 0, then stream speed is positive
! so sweep from more negative to more positive zed
i = llim
df(i) = (f(i)-fl)/del
do i = llim+1, ulim
df(i) = (f(i)-f(i-1))/del
end do
end if
end subroutine fd_cell_centres_zed
! cell_centres_zed takes f at z grid locations
! and returns f at cell centres
! (with possible offset due to upwinding)
subroutine cell_centres_zed (llim, f, upwnd, sgn, fl, fr, fc)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: upwnd
integer, intent (in) :: sgn
complex, intent (in) :: fl, fr
complex, dimension (llim:), intent (out) :: fc
integer :: i, ulim
ulim = size(f)+llim-1
if (sgn > 0) then
! if sgn > 0, then stream speed is negative
! so sweep from more positive to more negative zed
i = ulim
fc(i) = 0.5*((1.-upwnd)*fr + (1.+upwnd)*f(i))
do i = ulim-1, llim, -1
fc(i) = 0.5*((1.-upwnd)*f(i+1) + (1.+upwnd)*f(i))
end do
else
! if sgn < 0, then stream speed is positive
! so sweep from more negative to more positive zed
i = llim
fc(i) = 0.5*((1.+upwnd)*f(i)+(1.-upwnd)*fl)
do i = llim+1, ulim
fc(i) = 0.5*((1.+upwnd)*f(i)+(1.-upwnd)*f(i-1))
end do
end if
end subroutine cell_centres_zed
subroutine fd_variable_upwinding_zed (llim, iseg, nseg, f, del, sgn, upwnd, fl, fr, periodic, df)
implicit none
integer, intent (in) :: llim, iseg, nseg
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del, upwnd
integer, intent (in) :: sgn
complex, dimension (:), intent (in) :: fl, fr
logical, intent (in) :: periodic
complex, dimension (llim:), intent (out) :: df
integer :: i, istart, iend, ulim
! if upwnd is zero or if vpa=0, then use centered differences
if (abs(upwnd) < epsilon(0.) .or. sgn == 0) then
call second_order_centered_zed (llim, iseg, nseg, f, del, sgn, fl, fr, periodic, df)
else
ulim = size(f)+llim-1
! if sgn > 0, then stream speed is negative
! so sweep from more positive to more negative zed
if (sgn > 0) then
if (iseg == nseg.and..not.periodic) then
i = ulim
df(i) = (0.5*(upwnd-1.)*f(i-1)-upwnd*f(i))/del
else
i = ulim
df(i) = (0.5*(upwnd-1.)*f(i-1)-upwnd*f(i)+0.5*(1.+upwnd)*fr(1))/del
end if
if (iseg == 1.and..not.periodic) then
i = llim
! at left boundary, must upwind fully as no info for f(i-1)
df(i) = (f(i+1)-f(i))/del
else
i = llim
df(i) = (0.5*(1.+upwnd)*f(i+1)-upwnd*f(i)+0.5*(upwnd-1.)*fl(2))/del
end if
istart = ulim
iend = llim
else
if (iseg == 1.and..not.periodic) then
i = llim
df(i) = (0.5*(1.-upwnd)*f(i+1)+upwnd*f(i))/del
else
i = llim
df(i) = (0.5*(1.-upwnd)*f(i+1)+upwnd*f(i)-0.5*(1.+upwnd)*fl(2))/del
end if
if (iseg == nseg.and..not.periodic) then
i = ulim
! if at rightmost zed, have no info for f(i+1) so must fully upwind
df(i) = (f(i)-f(i-1))/del
else
i = ulim
df(i) = (0.5*(1.-upwnd)*fr(1)+upwnd*f(i)-0.5*(1.+upwnd)*f(i-1))/del
end if
istart = llim
iend = ulim
end if
! mixed 2nd order centered and 1st order upwind scheme
do i = istart-sgn, iend+sgn, -sgn
df(i) = sgn*(0.5*(1.+upwnd)*f(i+sgn) - upwnd*f(i) + 0.5*(upwnd-1.)*f(i-sgn))/del
end do
end if
end subroutine fd_variable_upwinding_zed
subroutine fd_variable_upwinding_vpa (llim, f, del, sgn, upwnd, df)
implicit none
integer, intent (in) :: llim
complex, dimension (llim:), intent (in) :: f
real, intent (in) :: del, upwnd
integer, intent (in) :: sgn
complex, dimension (llim:), intent (out) :: df
integer :: i, istart, iend, ulim
! if upwnd is zero or if z=0, then use centered differences
if (abs(upwnd) < epsilon(0.) .or. sgn == 0) then
call second_order_centered_vpa (llim, f, del, df)
else
ulim = size(f)+llim-1
! if sgn > 0, then stream speed is negative
! so sweep from more positive to more negative zed
if (sgn > 0) then
istart = ulim
iend = llim
else
istart = llim
iend = ulim
end if
! zero_bc assumes that g -> zero beyond grid
! boundaries in vpa
df(istart) = sgn*(0.5*(upwnd-1.0)*f(istart-sgn)-upwnd*f(istart))/del
! as do not have info beyond grid boundary at end of sweep
! use pure upwinding
df(iend) = sgn*(f(iend+sgn)-f(iend))/del
! mixed centered and 1st order upwind scheme
do i = istart-sgn, iend+sgn, -sgn
df(i) = sgn*(0.5*(1.+upwnd)*f(i+sgn) - upwnd*f(i) + 0.5*(upwnd-1.)*f(i-sgn))/del
end do
end if
end subroutine fd_variable_upwinding_vpa
! only good for equally-spaced grid-pts
subroutine fd3pt_real (prof, profgrad, dr)
implicit none
real, dimension (:), intent (in) :: prof
real, dimension (:), intent (out) :: profgrad
real, intent (in) :: dr
integer :: ix, npts
real, dimension (:), allocatable :: aa, bb, cc
npts = size(prof)
allocate (aa(npts), bb(npts), cc(npts))
aa = 1.0 ; bb = 4.0 ; cc = 1.0
aa(1) = 0.0 ; bb(1) = 0.5 ; cc(1) = 0.5
aa(npts) = 0.5 ; bb(npts) = 0.5 ; cc(npts) = 0.0
do ix = 2, npts-1
profgrad(ix) = 3.0 * (prof(ix+1) - prof(ix-1)) / dr
end do
profgrad(1) = (prof(2)-prof(1))/dr
profgrad(npts) = (prof(npts)-prof(npts-1))/dr
call tridag (aa, bb, cc, profgrad)
deallocate (aa, bb, cc)
end subroutine fd3pt_real
subroutine fd3pt_real_array (prof, profgrad, dr)
implicit none
real, dimension (:), intent (in) :: prof, dr
real, dimension (:), intent (out) :: profgrad
integer :: ix, npts
real :: a, b, c
npts = size(prof)
do ix = 2, npts-1
profgrad(ix) = ((prof(ix)-prof(ix-1))*dr(ix)/dr(ix-1) &
+ (prof(ix+1)-prof(ix))*dr(ix-1)/dr(ix)) / (dr(ix-1)+dr(ix))
end do
ix = 1
a = -(2.*dr(1) + dr(2))/(dr(1)*(dr(1)+dr(2)))
b = (dr(1)+dr(2))/(dr(1)*dr(2))
c = -dr(1)/(dr(2)*(dr(1)+dr(2)))
profgrad(1) = a*prof(1)+b*prof(2)+c*prof(3)
ix = npts
a = dr(npts-1)/(dr(npts-2)*(dr(npts-2)+dr(npts-1)))
b = -(dr(npts-1)+dr(npts-2))/(dr(npts-2)*dr(npts-1))
c = (2.*dr(npts-1)+dr(npts-2))/(dr(npts-1)*(dr(npts-1)+dr(npts-2)))
profgrad(npts) = a*prof(npts-2) + b*prof(npts-1) + c*prof(npts)
end subroutine fd3pt_real_array
subroutine fd3pt_complex_array (prof, profgrad, dr)
implicit none
complex, dimension (:), intent (in) :: prof
real, dimension (:), intent (in) :: dr
complex, dimension (:), intent (out) :: profgrad
integer :: ix, npts
real :: a, b, c
npts = size(prof)
do ix = 2, npts-1
profgrad(ix) = ((prof(ix)-prof(ix-1))*dr(ix)/dr(ix-1) &
+ (prof(ix+1)-prof(ix))*dr(ix-1)/dr(ix)) / (dr(ix-1)+dr(ix))
end do
ix = 1
a = -(2.*dr(1) + dr(2))/(dr(1)*(dr(1)+dr(2)))
b = (dr(1)+dr(2))/(dr(1)*dr(2))
c = -dr(1)/(dr(2)*(dr(1)+dr(2)))
profgrad(1) = a*prof(1)+b*prof(2)+c*prof(3)
ix = npts
a = dr(npts-1)/(dr(npts-2)*(dr(npts-2)+dr(npts-1)))
b = -(dr(npts-1)+dr(npts-2))/(dr(npts-2)*dr(npts-1))
c = (2.*dr(npts-1)+dr(npts-2))/(dr(npts-1)*(dr(npts-1)+dr(npts-2)))
profgrad(npts) = a*prof(npts-2) + b*prof(npts-1) + c*prof(npts)
end subroutine fd3pt_complex_array