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INTERPOLATIONS: Rewrite the monotonicity-preserving limiter.

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Restore the original Suresh & Huynh limiter and add modification by
Ahn & Lee for compact schemes. Additionally, pass the variable
positivity flag and for positive variables modifie the lower limit
using the minimum value obtained from the cubic spline interpolation.

Signed-off-by: Grzegorz Kowal <grzegorz@amuncode.org>
This commit is contained in:
Grzegorz Kowal 2023-02-07 10:11:48 -03:00
parent 81de98d9e2
commit 57795f0ca3
1 changed files with 152 additions and 106 deletions
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262
sources/interpolations.F90
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@ -42,7 +42,8 @@ module interpolations
real(kind=8), dimension(:,:,:) , intent(in) :: q
real(kind=8), dimension(:,:,:,:,:), intent(out) :: qi
end subroutine
subroutine reconstruct_iface(h, fc, fl, fr)
subroutine reconstruct_iface(positive, h, fc, fl, fr)
logical , intent(in) :: positive
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -76,7 +77,6 @@ module interpolations
! monotonicity preserving reconstruction coefficients
!
real(kind=8), save :: kappa = 1.0d+00
real(kind=8), save :: kbeta = 1.0d+00
! number of ghost zones (required for compact schemes)
!
@ -218,7 +218,6 @@ module interpolations
call get_parameter("eps" , eps )
call get_parameter("limo3_rad" , rad )
call get_parameter("kappa" , kappa )
call get_parameter("kbeta" , kbeta )
call get_parameter("ppm_const" , ppm_const )
call get_parameter("central_weight" , cweight )
call get_parameter("cfl" , cfl )
@ -978,20 +977,20 @@ module interpolations
!
! Arguments:
!
! positive - the variable positivity flag;
! p - the variable positivity flag;
! h - the spatial step;
! q - the variable array;
! qi - the array of reconstructed interfaces (2 in each direction);
!
!===============================================================================
!
subroutine interfaces_dir(positive, h, q, qi)
subroutine interfaces_dir(p, h, q, qi)
use coordinates, only : nb, ne, nbl, neu
implicit none
logical , intent(in) :: positive
logical , intent(in) :: p
real(kind=8), dimension(:) , intent(in) :: h
real(kind=8), dimension(:,:,:) , intent(in) :: q
real(kind=8), dimension(:,:,:,:,:), intent(out) :: qi
@ -1025,23 +1024,23 @@ module interpolations
do k = nbl, neu
#endif /* NDIMS == 3 */
do j = nbl, neu
call reconstruct(h(1), q(:,j,k), qi(:,j,k,1,1), qi(:,j,k,2,1))
call reconstruct(p, h(1), q(:,j,k), qi(:,j,k,1,1), qi(:,j,k,2,1))
end do ! j = nbl, neu
do i = nbl, neu
call reconstruct(h(2), q(i,:,k), qi(i,:,k,1,2), qi(i,:,k,2,2))
call reconstruct(p, h(2), q(i,:,k), qi(i,:,k,1,2), qi(i,:,k,2,2))
end do ! i = nbl, neu
#if NDIMS == 3
end do ! k = nbl, neu
do j = nbl, neu
do i = nbl, neu
call reconstruct(h(3), q(i,j,:), qi(i,j,:,1,3), qi(i,j,:,2,3))
call reconstruct(p, h(3), q(i,j,:), qi(i,j,:,1,3), qi(i,j,:,2,3))
end do ! i = nbl, neu
end do ! j = nbl, neu
#endif /* NDIMS == 3 */
! make sure the interface states are positive for positive variables
!
if (positive) then
if (p) then
#if NDIMS == 3
do k = nbl, neu
@ -1442,6 +1441,7 @@ module interpolations
!
! Arguments:
!
! p - the flag indicating if the reconstructed variable is positivite;
! h - the spatial step; this is required for some reconstruction methods;
! f - the input vector of cell averaged values;
! fl - the left side state reconstructed for location (i+1/2);
@ -1449,10 +1449,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct(h, f, fl, fr)
subroutine reconstruct(p, h, f, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: f
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -1461,7 +1462,7 @@ module interpolations
!
! reconstruct the states using the selected subroutine
!
call reconstruct_states(h, f(:), fl(:), fr(:))
call reconstruct_states(p, h, f(:), fl(:), fr(:))
! correct the reconstruction near extrema by clipping them in order to improve
! the stability of scheme
@ -1485,10 +1486,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_tvd(h, fc, fl, fr)
subroutine reconstruct_tvd(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -1553,10 +1555,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_weno3(h, fc, fl, fr)
subroutine reconstruct_weno3(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -1675,10 +1678,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_limo3(h, fc, fl, fr)
subroutine reconstruct_limo3(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -1818,10 +1822,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_ppm(h, fc, fl, fr)
subroutine reconstruct_ppm(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -2002,10 +2007,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_weno5z(h, fc, fl, fr)
subroutine reconstruct_weno5z(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -2159,10 +2165,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_weno5yc(h, fc, fl, fr)
subroutine reconstruct_weno5yc(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -2317,10 +2324,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_weno5ns(h, fc, fl, fr)
subroutine reconstruct_weno5ns(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -2503,12 +2511,13 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_crweno5z(h, fc, fl, fr)
subroutine reconstruct_crweno5z(p, h, fc, fl, fr)
use algebra, only : tridiag
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -2878,12 +2887,13 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_crweno5yc(h, fc, fl, fr)
subroutine reconstruct_crweno5yc(p, h, fc, fl, fr)
use algebra, only : tridiag
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -3256,12 +3266,13 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_crweno5ns(h, fc, fl, fr)
subroutine reconstruct_crweno5ns(p, h, fc, fl, fr)
use algebra, only : tridiag
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -3647,10 +3658,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_mp5(h, fc, fl, fr)
subroutine reconstruct_mp5(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -3676,7 +3688,7 @@ module interpolations
u(n-1) = sum(ce3(:) * fc(n-2:n))
u(n ) = fc(n )
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
fl(1:n) = u(1:n)
@ -3694,7 +3706,7 @@ module interpolations
u(n-1) = sum(ce3(:) * fi(n-2:n))
u(n ) = fi(n )
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
fr(1:n-1) = u(n-1:1:-1)
@ -3724,10 +3736,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_mp7(h, fc, fl, fr)
subroutine reconstruct_mp7(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -3755,7 +3768,7 @@ module interpolations
u(n-1) = sum(ce3(:) * fc(n-2:n))
u(n ) = fc(n )
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
fl(1:n) = u(1:n)
@ -3775,7 +3788,7 @@ module interpolations
u(n-1) = sum(ce3(:) * fi(n-2:n))
u(n ) = fi(n )
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
fr(1:n-1) = u(n-1:1:-1)
@ -3805,10 +3818,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_mp9(h, fc, fl, fr)
subroutine reconstruct_mp9(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -3838,7 +3852,7 @@ module interpolations
u(n-1) = sum(ce3(:) * fc(n-2:n))
u(n ) = fc(n )
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
fl(1:n) = u(1:n)
@ -3860,7 +3874,7 @@ module interpolations
u(n-1) = sum(ce3(:) * fi(n-2:n))
u(n ) = fi(n )
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
fr(1:n-1) = u(n-1:1:-1)
@ -3902,12 +3916,13 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_crmp5(h, fc, fl, fr)
subroutine reconstruct_crmp5(p, h, fc, fl, fr)
use algebra, only : tridiag
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -3958,7 +3973,7 @@ module interpolations
call tridiag(n, a(1:n), b(1:n), c(1:n), r(1:n), u(1:n))
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
fl(1:n) = u(1:n)
@ -3983,7 +3998,7 @@ module interpolations
call tridiag(n, a(1:n), b(1:n), c(1:n), r(1:n), u(1:n))
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
fr(1:n-1) = u(n-1:1:-1)
@ -4014,12 +4029,13 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_crmp7(h, fc, fl, fr)
subroutine reconstruct_crmp7(p, h, fc, fl, fr)
use algebra, only : tridiag
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -4072,7 +4088,7 @@ module interpolations
call tridiag(n, a(1:n), b(1:n), c(1:n), r(1:n), u(1:n))
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
fl(1:n) = u(1:n)
@ -4099,7 +4115,7 @@ module interpolations
call tridiag(n, a(1:n), b(1:n), c(1:n), r(1:n), u(1:n))
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
fr(1:n-1) = u(n-1:1:-1)
@ -4130,12 +4146,13 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_crmp9(h, fc, fl, fr)
subroutine reconstruct_crmp9(p, h, fc, fl, fr)
use algebra, only : pentadiag
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -4196,7 +4213,7 @@ module interpolations
call pentadiag(n, e(:), c(:), d(:), a(:), b(:), r(:), u(:))
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
fl(1:n) = u(1:n)
@ -4225,7 +4242,7 @@ module interpolations
call pentadiag(n, e(:), c(:), d(:), a(:), b(:), r(:), u(:))
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
fr(1:n-1) = u(n-1:1:-1)
@ -4260,12 +4277,13 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_ocmp5(h, fc, fl, fr)
subroutine reconstruct_ocmp5(p, h, fc, fl, fr)
use algebra, only : tridiag
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -4327,7 +4345,7 @@ module interpolations
call tridiag(n, a(1:n), b(1:n), c(1:n), r(1:n), u(1:n))
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
fl(1:n) = u(1:n)
@ -4352,7 +4370,7 @@ module interpolations
call tridiag(n, a(1:n), b(1:n), c(1:n), r(1:n), u(1:n))
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
fr(1:n-1) = u(n-1:1:-1)
@ -4387,12 +4405,13 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_ocmp7(h, fc, fl, fr)
subroutine reconstruct_ocmp7(p, h, fc, fl, fr)
use algebra, only : pentadiag
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -4466,7 +4485,7 @@ module interpolations
call pentadiag(n, e(:), c(:), d(:), a(:), b(:), r(:), u(:))
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
fl(1:n) = u(1:n)
@ -4493,7 +4512,7 @@ module interpolations
call pentadiag(n, e(:), c(:), d(:), a(:), b(:), r(:), u(:))
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
fr(1:n-1) = u(n-1:1:-1)
@ -4528,12 +4547,13 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_ocmp9(h, fc, fl, fr)
subroutine reconstruct_ocmp9(p, h, fc, fl, fr)
use algebra, only : pentadiag
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -4611,7 +4631,7 @@ module interpolations
call pentadiag(n, e(:), c(:), d(:), a(:), b(:), r(:), u(:))
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
fl(1:n) = u(1:n)
@ -4640,7 +4660,7 @@ module interpolations
call pentadiag(n, e(:), c(:), d(:), a(:), b(:), r(:), u(:))
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
fr(1:n-1) = u(n-1:1:-1)
@ -4773,10 +4793,11 @@ module interpolations
!
!===============================================================================
!
subroutine reconstruct_gp(h, fc, fl, fr)
subroutine reconstruct_gp(p, h, fc, fl, fr)
implicit none
logical , intent(in) :: p
real(kind=8) , intent(in) :: h
real(kind=8), dimension(:), intent(in) :: fc
real(kind=8), dimension(:), intent(out) :: fl, fr
@ -4814,7 +4835,7 @@ module interpolations
! apply the monotonicity preserving limiting
!
call mp_limiting(fc(:), u(:))
call mp_limiting(p, n, fc(:), u(:))
! copy the interpolation to the respective vector
!
@ -4848,7 +4869,7 @@ module interpolations
! apply the monotonicity preserving limiting
!
call mp_limiting(fi(:), u(:))
call mp_limiting(p, n, fi(:), u(:))
! copy the interpolation to the respective vector
!
@ -5325,93 +5346,118 @@ module interpolations
! subroutine MP_LIMITING:
! ----------------------
!
! Subroutine applies the monotonicity preserving (MP) limiter to a vector of
! high order reconstructed interface values.
! Subroutine applies the monotonicity preserving (MP) limitation to
! the interface states reconstructed using a high order upwind
! interpolation.
!
! Arguments:
!
! qc - the vector of cell-centered values;
! qi - the vector of interface values obtained from the high order
! interpolation as input and its monotonicity limited values as output;
! p - the logical flag indicating if the variable is positive;
! n - the length of vector u and q;
! u - the vector of centered cell-integrated values;
! q - the vector of interface values obtained from the high order
! interpolation to which the limitation is applied;
!
! References:
!
! [1] Suresh, A. & Huynh, H. T.,
! "Accurate Monotonicity-Preserving Schemes with Runge-Kutta
! Time Stepping"
! Journal on Computational Physics,
! 1997, vol. 136, pp. 83-99,
! http://dx.doi.org/10.1006/jcph.1997.5745
! [2] He, ZhiWei, Li, XinLiang, Fu, DeXun, & Ma, YanWen,
! "A 5th order monotonicity-preserving upwind compact difference
! scheme",
! Science China Physics, Mechanics and Astronomy,
! Volume 54, Issue 3, pp. 511-522,
! http://dx.doi.org/10.1007/s11433-010-4220-x
! Journal on Computational Physics, 1997, 136, 83-99,
! https://doi.org/10.1006/jcph.1997.5745
! [2] Myeong-Hwan Ahn1, Duck-Joo Lee,
! "Modified Monotonicity Preserving Constraints for High-Resolution
! Optimized Compact Scheme",
! Journal of Scientific Computing, 2020, 83:34,
! https://doi.org/10.1007/s10915-020-01221-0
!
!===============================================================================
!
subroutine mp_limiting(qc, qi)
subroutine mp_limiting(p, n, u, q)
implicit none
real(kind=8), dimension(:), intent(in) :: qc
real(kind=8), dimension(:), intent(inout) :: qi
logical , intent(in) :: p
integer , intent(in) :: n
real(kind=8), dimension(n), intent(in) :: u
real(kind=8), dimension(n), intent(inout) :: q
integer :: n, i, im1, ip1, ip2
real(kind=8) :: dq, ds, dc0, dc4, dm1, dp1, dml, dmr
integer :: i, im2, im1, ip1, ip2
real(kind=8) :: du, dm, dc, dp, dc4, dmm, dmp, di, de, bt
real(kind=8) :: qlc, qmd, qmp, qmn, qmx, qul
real(kind=8), dimension(0:size(qc)+2) :: dm
real(kind=8), dimension(n) :: d
real(kind=8), dimension(2) :: umn, umx
!-------------------------------------------------------------------------------
!
n = size(qc)
d(1 ) = 0.0d+00
d(2:n) = u(2:n) - u(1:n-1)
dm(0 ) = 0.0d+00
dm(1 ) = 0.0d+00
dm(2:n) = qc(2:n) - qc(1:n-1)
dm(n+1) = 0.0d+00
dm(n+2) = 0.0d+00
im2 = 1
im1 = 1
ip1 = 2
ip2 = 3
do i = 1, n - 1
do i = 1, n
ip1 = i + 1
du = kappa * d(i)
if (dm(i) * dm(ip1) >= 0.0d+00) then
dq = kappa * dm(i)
qmp = u(i) + minmod(d(ip1), du)
if ((q(i) - u(i)) * (q(i) - qmp) > eps) then
dm = d(i ) - d(im1)
dc = d(ip1) - d(i )
dp = d(ip2) - d(ip1)
dc4 = 4.0d+00 * dc
dmp = minmod4(dc4 - dp, 4.0d+00 * dp - dc, dc, dp)
dmm = minmod4(dc4 - dm, 4.0d+00 * dm - dc, dc, dm)
qul = u(i) + du
qmd = u(i) + 5.0d-01 * (d(ip1) - dmp )
qlc = u(i) + 5.0d-01 * (d(i ) + 8.0d+00 * dmm / 3.0d+00)
umn(1) = min(u(im1), u(ip1))
umn(2) = min(u(im2), u(ip2))
umx(1) = max(u(im1), u(ip1))
umx(2) = max(u(im2), u(ip2))
if ((u(i) > umx(1) .and. umx(1) > umx(2) .and. &
u(im2) <= u(ip1) .and. u(im1) >= u(ip2)) .or. &
(u(i) < umn(1) .and. umn(1) < umn(2) .and. &
u(im2) >= u(ip1) .and. u(im1) <= u(ip2))) then
bt = 2.0d+00 * u(i)
di = (u(im1) + u(ip1)) - bt
de = (u(im2) + u(ip2)) - bt
bt = di / de
if (bt > 3.0d-01) qlc = u(im2)
end if
qmx = max(min(u(i), u(ip1), qmd), min(u(i), qul, qlc))
qmn = min(max(u(i), u(ip1), qmd), max(u(i), qul, qlc))
if (p .and. qmn <= 0.0d+00) then
if (d(i) <= 0.0d+00 .and. d(ip1) >= 0.0d+00) then
qmn = u(i) * u(ip1) / (u(i) + u(ip1))
else
dq = kbeta * dm(i)
qmn = max(eps, min(u(i), u(ip1)))
end if
end if
qmp = qc(i) + minmod(dm(ip1), dq)
ds = (qi(i) - qc(i)) * (qi(i) - qmp)
if (ds > eps) then
im1 = i - 1
ip2 = i + 2
dm1 = dm(i ) - dm(im1)
dc0 = dm(ip1) - dm(i )
dp1 = dm(ip2) - dm(ip1)
dc4 = 4.0d+00 * dc0
dml = 0.5d+00 * minmod4(dc4 - dm1, 4.0d+00 * dm1 - dc0, dc0, dm1)
dmr = 0.5d+00 * minmod4(dc4 - dp1, 4.0d+00 * dp1 - dc0, dc0, dp1)
qmd = qc(i) + 0.5d+00 * dm(ip1) - dmr
qul = qc(i) + dq
qlc = qc(i) + 0.5d+00 * dq + dml
qmx = max(min(qc(i), qc(ip1), qmd), min(qc(i), qul, qlc))
qmn = min(max(qc(i), qc(ip1), qmd), max(qc(i), qul, qlc))
qi(i) = median(qi(i), qmn, qmx)
q(i) = median(q(i), qmn, qmx)
end if
end do ! i = 1, n - 1
im2 = im1
im1 = i
ip1 = ip2
ip2 = min(ip2 + 1, n)
end do
!-------------------------------------------------------------------------------
!