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|
/*@@
@file GRHydro_PPM.F90
@date Sun Feb 10 16:53:29 2002
@author Ian Hawke, Toni Font, Luca Baiotti, Frank Loeffler
@desc
Routines to do PPM reconstruction.
@enddesc
@@*/
#include "cctk.h"
#include "cctk_Arguments.h"
#include "cctk_Parameters.h"
#include "GRHydro_Macros.h"
subroutine PPM_TVD(origm, orig, origp, bextm, bextp)
CCTK_REAL :: origm, orig, origp, bextm, bextp
CCTK_REAL :: dloc, dupw, delta
dupw = orig - origm
dloc = origp - orig
if (dupw*dloc < 0.d0) then
delta=0.d0
else if (abs(dupw) < abs(dloc)) then
delta=dupw
else
delta=dloc
end if
bextm = orig - 0.5d0 * delta
bextp = orig + 0.5d0 * delta
end subroutine PPM_TVD
/*@@
@routine SimplePPM_1d
@date Thu Feb 14 19:08:52 2002
@author Ian Hawke, Toni Font, Christian Reisswig
@desc
The simple PPM reconstruction routine that applies along
each one dimensional slice.
@enddesc
@calls
@calledby
@history
Written in frustration when IH couldn''t get Toni''s original code
to work.
Later extended to enhanced PPM scheme by CR.
@endhistory
@@*/
#define SpaceMask_CheckStateBitsF90_1D(mask,i,type_bits,state_bits) \
(iand(mask((i)),(type_bits)).eq.(state_bits))
subroutine SimplePPM_1d(handle,poly,&
nx,dx,rho,velx,vely,velz,eps,press,rhominus,&
velxminus,velyminus,velzminus,epsminus,rhoplus,velxplus,velyplus,&
velzplus,epsplus,trivial_rp, hydro_excision_mask,&
gxx, gxy, gxz, gyy, gyz, gzz, beta, alp, w_lorentz, &
dir, ni, nj, nrx, nry, nrz, ev_l, ev_r, xw, rho_min)
USE GRHydro_Scalars
USE GRHydro_Eigenproblem
implicit none
DECLARE_CCTK_PARAMETERS
CCTK_INT :: handle,poly,nx
CCTK_REAL :: dx
CCTK_REAL, dimension(nx) :: rho,velx,vely,velz,eps
CCTK_REAL, dimension(nx) :: rhominus,velxminus,velyminus,velzminus,epsminus
CCTK_REAL, dimension(nx) :: rhoplus,velxplus,velyplus,velzplus,epsplus
CCTK_REAL, dimension(nx) :: rhominusl,velxminusl,velyminusl,velzminusl
CCTK_REAL, dimension(nx) :: epsminusl
CCTK_REAL, dimension(nx) :: rhoplusl,velxplusl,velyplusl,velzplusl,epsplusl
CCTK_REAL, dimension(nx) :: rhominusr,velxminusr,velyminusr,velzminusr
CCTK_REAL, dimension(nx) :: epsminusr
CCTK_REAL, dimension(nx) :: rhoplusr,velxplusr,velyplusr,velzplusr,epsplusr
CCTK_REAL, dimension(nx) :: atmosphere_mask
CCTK_INT :: i,s
CCTK_REAL, dimension(nx) :: drho,dvelx,dvely,dvelz,deps
CCTK_REAL, dimension(nx) :: dmrho,dmvelx,dmvely,dmvelz,dmeps
CCTK_REAL, dimension(nx) :: press,dpress,d2rho,tilde_flatten
CCTK_REAL :: dpress2,dvel,w,flatten,eta,etatilde
logical, dimension(nx) :: trivial_rp
CCTK_INT, dimension(nx) :: hydro_excision_mask
CCTK_REAL, dimension(nx) :: gxx, gxy, gxz, gyy, gyz, gzz, &
beta, alp, w_lorentz
CCTK_INT :: dir, ni, nj, nrx, nry, nrz
CCTK_REAL, dimension(nrx, nry, nrz) :: ev_l, ev_r, xw
CCTK_REAL :: uxx, uxy, uxz, uyy, uyz, uzz, det
CCTK_REAL, dimension(5) :: lam
CCTK_REAL :: dupw, dloc, delta
CCTK_REAL :: agxx, agxy, agxz, agyy, agyz, agzz
CCTK_REAL, dimension(nx) :: xwind, l_ev_l, l_ev_r
!CCTK_INT, dimension(nx) :: use_std_ppm
CCTK_REAL :: D2a, D2aL, D2aR, D2aC, D2aLim, rhi, threshold, rho_min, daplus, daminus, D3a, D3aLL, D3aL, D3aR, D2aLL, D2aRR, D3aMin, D3aMax
logical :: cond
!!$ Initially, all the Riemann problems will be trivial
trivial_rp = .true.
!use_std_ppm = 1
#define STEEP(x,dx,dmx) \
if ( (x(i+1) - x(i)) * (x(i) - x(i-1)) > 0.d0 ) then &&\
dmx(i) = sign(1.d0, dx(i)) * \
min(abs(dx(i)), 2.d0 * abs(x(i) - x(i-1)), \
2.d0 * abs(x(i+1) - x(i))) &&\
else &&\
dmx(i) = 0.d0 &&\
end if
if (use_enhanced_ppm .eq. 0) then
!! This is the original PPM algorithm by Colella & Woodward 1984.
!!$ Average slopes delta_m(a). See (1.7) of Colella and Woodward, p.178
!!$ This is the expression for an even grid.
do i = 2, nx - 1
drho(i) = 0.5d0 * (rho(i+1) - rho(i-1))
dvelx(i) = 0.5d0 * (velx(i+1) - velx(i-1))
dvely(i) = 0.5d0 * (vely(i+1) - vely(i-1))
dvelz(i) = 0.5d0 * (velz(i+1) - velz(i-1))
dpress(i) = press(i+1) - press(i-1)
d2rho(i) = (rho(i+1) - 2.d0 * rho(i) + rho(i-1))! / 6.d0 / dx / dx
! since we use d2rho only for the condition d2rho(i+1)*d2rhoi(i-1)<0
! the denominator is not necessary
end do
if (poly .eq. 0) then
do i = 2, nx - 1
deps(i) = 0.5d0 * (eps(i+1) - eps(i-1))
end do
end if
!!$ Steepened slope. See (1.8) of Colella and Woodward, p.178
do i = 2, nx - 1
STEEP(rho, drho, dmrho)
STEEP(velx, dvelx, dmvelx)
STEEP(vely, dvely, dmvely)
STEEP(velz, dvelz, dmvelz)
end do
if (poly .eq. 0) then
do i = 2, nx - 1
STEEP(eps, deps, dmeps)
end do
end if
!!$ Initial boundary states. See (1.9) of Colella and Woodward, p.178
do i = 2, nx-2
rhoplus(i) = 0.5d0 * (rho(i) + rho(i+1)) + &
(dmrho(i) - dmrho(i+1)) / 6.d0
rhominus(i+1) = rhoplus(i)
velxplus(i) = 0.5d0 * (velx(i) + velx(i+1)) + &
(dmvelx(i) - dmvelx(i+1)) / 6.d0
velxminus(i+1) = velxplus(i)
velyplus(i) = 0.5d0 * (vely(i) + vely(i+1)) + &
(dmvely(i) - dmvely(i+1)) / 6.d0
velyminus(i+1) = velyplus(i)
velzplus(i) = 0.5d0 * (velz(i) + velz(i+1)) + &
(dmvelz(i) - dmvelz(i+1)) / 6.d0
velzminus(i+1) = velzplus(i)
end do
if (poly .eq. 0) then
do i = 2, nx-2
epsplus(i) = 0.5d0 * (eps(i) + eps(i+1)) + &
(dmeps(i) - dmeps(i+1)) / 6.d0
epsminus(i+1) = epsplus(i)
end do
end if
else
!! This is the modified PPM algorithm by Colella & Sekora 2008 and McCorquodale & Colella 2011.
!! This uses a better limiter based on second derivatives that preserves
!! accuracy at local extrema. It also uses a higher-order interpolation polynomial.
!!$ Initial boundary states (sixth order accurate). See (17) of Colella and Sekora 2008, p.7071
#define APPROX_AT_CELL_INTERFACE_STENCIL4(a, ah) \
ah = 37.0d0/60.0d0*(a(i)+a(i+1)) - 2.0d0/15.0d0*(a(i-1)+a(i+2)) + 1.0d0/60.0d0*(a(i-2)+a(i+3))
!!$ Initial boundary states (4th order accurate). See (16) of Colella and Sekora 2008, p.7071
#define APPROX_AT_CELL_INTERFACE(a, ah) \
ah = 7.0d0/12.0d0*(a(i)+a(i+1)) - 1.0d0/12.0d0*(a(i-1)+a(i+2))
!! A threshold value of rho
threshold = rho_min * GRHydro_ppm_atmo_tolerance
! .and. rho(i-2) .gt. threshold .and. rho(i+3) .gt. threshold
#define LIMIT(a,ah,C, alim) \
if ((min(a(i),a(i+1)) .le. ah) .and. (ah .le. max(a(i),a(i+1)))) then &&\
alim = ah &&\
else &&\
D2a = 3.0d0 * (a(i) - 2.0d0*ah + a(i+1)) &&\
D2aL = (a(i-1) - 2.0d0*a(i) + a(i+1)) &&\
D2aR = (a(i) - 2.0d0*a(i+1) + a(i+2)) &&\
D2aLim = sign(1.0d0, D2a)*min(C*abs(D2aL), C*abs(D2aR), abs(D2a))/3.0d0 &&\
if (D2a*D2aR .ge. 0 .and. D2a*D2aL .ge. 0 .and. abs(D2aLim) .lt. abs(0.5d0*(a(i)+a(i+1)))) then &&\
alim = 0.5d0*(a(i)+a(i+1)) - D2aLim &&\
else &&\
alim = 0.5d0*(a(i)+a(i+1)) &&\
end if &&\
endif
! .and. abs(D2aLim) .lt. abs(0.5d0*(a(i)+a(i+1)))
! .and. abs(D2aLim) .lt. abs(0.5d0*(a(i)+a(i+1)))
!sign(1.0d0, D2a)*min(abs(D2aLim), abs(0.5*(a(i)+a(i+1))))
if (PPM3) then
!! We initialize "plus" \equiv a_j+1/2 with (16) via APPROX_AT_CELL_INTERFACE,
!! then checking for (13) of Colella & Sekora 2008 and applying
!! (18) and (19) if (13) is not satisfied. This is done with LIMIT.
!! There, we also switch to lower order if we are near atmosphere values.
do i = 2, nx-2
APPROX_AT_CELL_INTERFACE(rho, rhoplus(i))
LIMIT(rho, rhoplus(i), enhanced_ppm_C2, rhoplus(i))
rhominus(i+1) = rhoplus(i)
APPROX_AT_CELL_INTERFACE(velx, velxplus(i))
LIMIT(velx, velxplus(i), enhanced_ppm_C2, velxplus(i))
velxminus(i+1) = velxplus(i)
APPROX_AT_CELL_INTERFACE(vely, velyplus(i))
LIMIT(vely, velyplus(i), enhanced_ppm_C2, velyplus(i))
velyminus(i+1) = velyplus(i)
APPROX_AT_CELL_INTERFACE(velz, velzplus(i))
LIMIT(velz, velzplus(i), enhanced_ppm_C2, velzplus(i))
velzminus(i+1) = velzplus(i)
end do
if (poly .eq. 0) then
do i = 2, nx-2
APPROX_AT_CELL_INTERFACE(eps, epsplus(i))
LIMIT(eps, epsplus(i), enhanced_ppm_C2, epsplus(i))
epsminus(i+1) = epsplus(i)
end do
end if
else
!! Same as above but for 4 stencil points using (17) of Colella & Sekora 2008 as
!! initial states.
do i = 3, nx-3
APPROX_AT_CELL_INTERFACE_STENCIL4(rho, rhoplus(i))
LIMIT(rho, rhoplus(i), enhanced_ppm_C2, rhoplus(i))
rhominus(i+1) = rhoplus(i)
APPROX_AT_CELL_INTERFACE_STENCIL4(velx, velxplus(i))
LIMIT(velx, velxplus(i), enhanced_ppm_C2, velxplus(i))
velxminus(i+1) = velxplus(i)
APPROX_AT_CELL_INTERFACE_STENCIL4(vely, velyplus(i))
LIMIT(vely, velyplus(i), enhanced_ppm_C2, velyplus(i))
velyminus(i+1) = velyplus(i)
APPROX_AT_CELL_INTERFACE_STENCIL4(velz, velzplus(i))
LIMIT(velz, velzplus(i), enhanced_ppm_C2, velzplus(i))
velzminus(i+1) = velzplus(i)
end do
if (poly .eq. 0) then
do i = 3, nx-3
APPROX_AT_CELL_INTERFACE_STENCIL4(eps, epsplus(i))
LIMIT(eps, epsplus(i), enhanced_ppm_C2, epsplus(i))
epsminus(i+1) = epsplus(i)
end do
end if
endif
!! Finally compute pressure gradient needed for flattening and shock detection
do i = 2, nx-1
dpress(i) = press(i+1) - press(i-1)
end do
endif
!!$Discontinuity steepening. See (1.14-17) of C&W.
!!$This is the detect routine which mat be activated with the ppm_detect parameter
!!$Note that this part really also depends on the grid being even.
!!$Note also that we don''t have access to the gas constant gamma.
!!$So this is just dropped from eq. (3.2) of C&W.
!!$We can get around this by just rescaling the constant k0 (ppm_k0 here).
if (use_enhanced_ppm .eq. 0) then
! Only for 1984 PPM scheme!
if (ppm_detect .ne. 0) then
if (use_enhanced_ppm .eq. 1) then
! make sure drho, d2rho and dmrho are computed everywhere!
do i = 2, nx-1
drho(i) = 0.5d0 * (rho(i+1) - rho(i-1))
d2rho(i) = (rho(i+1) - 2.d0 * rho(i) + rho(i-1))
end do
do i = 2, nx-1
STEEP(rho, drho, dmrho)
end do
endif
do i = 3, nx - 2
if ( (d2rho(i+1)*d2rho(i-1) < 0.d0).and.(abs(rho(i+1)-rho(i-1)) - &
ppm_epsilon_shock * min(abs(rho(i+1)), abs(rho(i-1))) > 0.d0) ) then
etatilde = (rho(i-2) - rho(i+2) + 4.d0 * drho(i)) / (drho(i) * 12.d0)
else
etatilde = 0.d0
end if
eta = max(0.d0, min(1.d0, ppm_eta1 * (etatilde - ppm_eta2)))
if (ppm_k0 * abs(drho(i)) * min(press(i-1),press(i+1)) < &
abs(dpress(i)) * min(rho(i-1), rho(i+1))) then
eta = 0.d0
end if
if (eta > 0.d0) then
trivial_rp(i-1) = .false.
trivial_rp(i) = .false.
end if
rhominus(i) = rhominus(i) * (1.d0 - eta) + &
(rho(i-1) + 0.5d0 * dmrho(i-1)) * eta
rhoplus(i) = rhoplus(i) * (1.d0 - eta) + &
(rho(i+1) - 0.5d0 * dmrho(i+1)) * eta
end do
end if
!!$ mppm
#define D_UPW(x) (0.5d0 * (x(i) + x(i+1)))
#define LEFT1(x) (13.d0*x(i+1)-5.d0*x(i+2)+x(i+3)+3.d0*x(i ))/12.d0
#define RIGHT1(x) (13.d0*x(i )-5.d0*x(i-1)+x(i-2)+3.d0*x(i+1))/12.d0
if (ppm_mppm .gt. 0) then
l_ev_l=0.d0
l_ev_r=0.d0
xwind=0.d0
do i=3, nx - 3
agxx = 0.5d0*( gxx(i) + gxx(i+1) )
agxy = 0.5d0*( gxy(i) + gxy(i+1) )
agxz = 0.5d0*( gxz(i) + gxz(i+1) )
agyy = 0.5d0*( gyy(i) + gyy(i+1) )
agyz = 0.5d0*( gyz(i) + gyz(i+1) )
agzz = 0.5d0*( gzz(i) + gzz(i+1) )
det = SPATIAL_DETERMINANT(agxx, agxy, agxz, \
agyy, agyz, agzz)
call UpperMetric (uxx, uxy, uxz, uyy, uyz, uzz, &
det, agxx, agxy, agxz, agyy, agyz, agzz)
call eigenvalues(handle,&
D_UPW(rho), D_UPW(velx), D_UPW(vely), D_UPW(velz), &
D_UPW(eps), D_UPW(w_lorentz), lam, &
agxx, agxy, agxz, agyy, agyz, agzz, &
uxx, D_UPW(alp), D_UPW(beta))
l_ev_l(i)=lam(1)
l_ev_r(i)=lam(5)
xwind(i) = (lam(1) + lam(5)) / (abs(lam(1)) + abs(lam(5)))
xwind(i) = min(1.d0, max(-1.d0, xwind(i)))
#define LEFTPLUS(x,xplus) xplus(i) = abs(xwind(i)) * LEFT1(x) + \
(1.d0-abs(xwind(i))) * xplus(i)
#define LEFTMINUS(x,xminus) xminus(i+1)= abs(xwind(i)) * LEFT1(x) + \
(1.d0-abs(xwind(i))) * xminus(i+1)
#define RIGHTPLUS(x,xplus) xplus(i) = abs(xwind(i)) * RIGHT1(x) + \
(1.d0-abs(xwind(i))) * xplus(i)
#define RIGHTMINUS(x,xminus) xminus(i+1)= abs(xwind(i)) * RIGHT1(x) + \
(1.d0-abs(xwind(i))) * xminus(i+1)
#define CHECK(x,xc) if (x(i+1) .gt. x(i)) then && xc=min(x(i+1),max(x(i),xc)) && else && xc=min(x(i),max(x(i+1),xc)) && endif
!!$ xwind(i)=0.d0
if (xwind(i) .lt. 0.0d0) then
LEFTPLUS(rho, rhoplus)
LEFTMINUS(rho, rhominus)
LEFTPLUS(velx, velxplus)
LEFTMINUS(velx, velxminus)
LEFTPLUS(vely, velyplus)
LEFTMINUS(vely, velyminus)
LEFTPLUS(velz, velzplus)
LEFTMINUS(velz, velzminus)
if (poly .eq. 0) then
LEFTPLUS(eps, epsplus)
LEFTMINUS(eps, epsminus)
end if
else
RIGHTPLUS(rho, rhoplus)
RIGHTMINUS(rho, rhominus)
RIGHTPLUS(velx, velxplus)
RIGHTMINUS(velx, velxminus)
RIGHTPLUS(vely, velyplus)
RIGHTMINUS(vely, velyminus)
RIGHTPLUS(velz, velzplus)
RIGHTMINUS(velz, velzminus)
if (poly .eq. 0) then
RIGHTPLUS(eps, epsplus)
RIGHTMINUS(eps, epsminus)
end if
end if
CHECK(rho, rhoplus(i))
CHECK(rho, rhominus(i+1))
CHECK(velx, velxplus(i))
CHECK(velx, velxminus(i+1))
CHECK(vely, velyplus(i))
CHECK(vely, velyminus(i+1))
CHECK(velz, velzplus(i))
CHECK(velz, velzminus(i+1))
if (poly .eq. 0) then
CHECK(eps, epsplus(i))
CHECK(eps, epsminus(i+1))
end if
!!$ if ((dir .eq. 1) .and. (ni .eq. 4) .and. (nj .eq. 4)) then
!!$ write (*,*) rhoplus(i), rhominus(i+1)
!!$ end if
end do
!!$ mppm debug output
if (ppm_mppm_debug_eigenvalues .gt. 0) then
if (dir .eq. 1) then
ev_l(:,ni,nj) = l_ev_l
ev_r(:,ni,nj) = l_ev_r
xw(:,ni,nj) = xwind
else if (dir .eq. 2) then
ev_l(ni,:,nj) = l_ev_l
ev_r(ni,:,nj) = l_ev_r
xw(ni,:,nj) = xwind
else if (dir .eq. 3) then
ev_l(ni,nj,:) = l_ev_l
ev_r(ni,nj,:) = l_ev_r
xw(ni,nj,:) = xwind
else
write (*,*) "flux direction not 1 to 3 ?"
end if
end if
end if
end if
!!$ Zone flattening. See appendix of C&W, p. 197-8.
do i = 3, nx - 2
dpress2 = press(i+2) - press(i-2)
dvel = velx(i+1) - velx(i-1)
if ( (abs(dpress(i)) > ppm_epsilon * min(press(i-1),press(i+1))) .and. &
(dvel < 0.d0) ) then
w = 1.d0
else
w = 0.d0
end if
if (abs(dpress2) < ppm_small) then
tilde_flatten(i) = 1.d0
else
tilde_flatten(i) = max(0.d0, 1.d0 - w * max(0.d0, ppm_omega2 * &
(dpress(i) / dpress2 - ppm_omega1)))
end if
end do
if (use_enhanced_ppm .eq. 0) then
! In 1984 PPM, flattening is applied before constraining parabolic profiles.
if (PPM3) then !!$ Implement C&W, page 197, but with a workaround which allows to use stencil=3.
do i = 3, nx - 2
flatten = tilde_flatten(i)
if (abs(1.d0 - flatten) > 0.d0) then
trivial_rp(i-1) = .false.
trivial_rp(i) = .false.
end if
rhoplus(i) = flatten * rhoplus(i) + (1.d0 - flatten) * rho(i)
rhominus(i) = flatten * rhominus(i) + (1.d0 - flatten) * rho(i)
velxplus(i) = flatten * velxplus(i) + (1.d0 - flatten) * velx(i)
velxminus(i) = flatten * velxminus(i) + (1.d0 - flatten) * velx(i)
velyplus(i) = flatten * velyplus(i) + (1.d0 - flatten) * vely(i)
velyminus(i) = flatten * velyminus(i) + (1.d0 - flatten) * vely(i)
velzplus(i) = flatten * velzplus(i) + (1.d0 - flatten) * velz(i)
velzminus(i) = flatten * velzminus(i) + (1.d0 - flatten) * velz(i)
if (poly .eq. 0) then
epsplus(i) = flatten * epsplus(i) + (1.d0 - flatten) * eps(i)
epsminus(i) = flatten * epsminus(i) + (1.d0 - flatten) * eps(i)
end if
end do
else !!$ Really implement C&W, page 197; which requires stencil 4.
do i = 4, nx - 3
s=sign(1.d0, -dpress(i))
flatten = max(tilde_flatten(i), tilde_flatten(i+s))
if (abs(1.d0 - flatten) > 0.d0) then
trivial_rp(i-1) = .false.
trivial_rp(i) = .false.
end if
rhoplus(i) = flatten * rhoplus(i) + (1.d0 - flatten) * rho(i)
rhominus(i) = flatten * rhominus(i) + (1.d0 - flatten) * rho(i)
velxplus(i) = flatten * velxplus(i) + (1.d0 - flatten) * velx(i)
velxminus(i) = flatten * velxminus(i) + (1.d0 - flatten) * velx(i)
velyplus(i) = flatten * velyplus(i) + (1.d0 - flatten) * vely(i)
velyminus(i) = flatten * velyminus(i) + (1.d0 - flatten) * vely(i)
velzplus(i) = flatten * velzplus(i) + (1.d0 - flatten) * velz(i)
velzminus(i) = flatten * velzminus(i) + (1.d0 - flatten) * velz(i)
if (poly .eq. 0) then
epsplus(i) = flatten * epsplus(i) + (1.d0 - flatten) * eps(i)
epsminus(i) = flatten * epsminus(i) + (1.d0 - flatten) * eps(i)
end if
end do
end if
endif
!!$ Monotonicity. See (1.10) of C&W.
#define MON(xminus,x,xplus) \
if (.not.( (xplus(i).eq.x(i)) .and. (x(i).eq.xminus(i)) ) \
.and. ((xplus(i)-x(i))*(x(i)-xminus(i)) .le. 0.d0)) then&&\
trivial_rp(i-1) = .false. &&\
trivial_rp(i) = .false. &&\
xminus(i) = x(i) &&\
xplus(i) = x(i) &&\
else if (6.d0 * (xplus(i) - xminus(i)) * (x(i) - 0.5d0 * \
(xplus(i) + xminus(i))) > \
(xplus(i) - xminus(i))**2) then &&\
xminus(i) = 3.d0 * x(i) - 2.d0 * xplus(i) &&\
trivial_rp(i-1) = .false. &&\
trivial_rp(i) = .false. &&\
else if (6.d0 * (xplus(i) - xminus(i)) * (x(i) - 0.5d0 * \
(xplus(i) + xminus(i))) < \
-(xplus(i) - xminus(i))**2) then &&\
xplus(i) = 3.d0 * x(i) - 2.d0 * xminus(i) &&\
trivial_rp(i-1) = .false. &&\
trivial_rp(i) = .false. &&\
end if &&\
if (.not.( (xplus(i).eq.x(i)) .and. (x(i).eq.xminus(i)) ) ) then &&\
trivial_rp(i-1) = .false. &&\
trivial_rp(i) = .false. &&\
end if
!! Monotonicity of PPM 2011 (McCorquodale & Colella 2011), Sec. 2.4.1, Eq. 23-34
!! This requires 4 stencil points.
#define MON_WITH_LOCAL_EXTREMUM_STENCIL4(aminus, a, aplus, C, C3) \
daplus = aplus(i)-a(i) &&\
daminus = a(i)-aminus(i) &&\
if (daplus*daminus .le. 0 .or. (a(i-2)-a(i))*(a(i)-a(i+2)) .le. 0) then &&\
D2a = - (12.0d0*a(i) - 6.0d0*(aminus(i)+aplus(i))) && \
D2aC = a(i-1) - 2.0d0*a(i) + a(i+1) &&\
D2aL = a(i-2) - 2.0d0*a(i-1) + a(i) &&\
D2aLL = a(i-3) - 2.0d0*a(i-2) + a(i-1) &&\
D2aR = a(i) - 2.0d0*a(i+1) + a(i+2) &&\
D2aRR = a(i+1) - 2.0d0*a(i+2) + a(i+3) &&\
D3a = D2aR - D2aC &&\
D3aL = D2aC - D2aL &&\
D3aR = D2aRR - D2aR &&\
D3aLL = D2aL - D2aLL &&\
if (sign(1.0d0, D2a) .eq. sign(1.0d0, D2aC) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aL) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aR)) then &&\
D2aLim = sign(1.0d0, D2a) * min(C*abs(D2aL), C*abs(D2aR), C*abs(D2aC), abs(D2a)) &&\
else &&\
D2aLim = 0 &&\
end if &&\
if (abs(D2a) .le. 1.d-12*max(abs(a(i-2)), abs(a(i-1)), abs(a(i)), abs(a(i+1)), abs(a(i+2)))) then &&\
rhi = 0 &&\
else &&\
rhi = D2aLim / D2a &&\
endif &&\
if (.not. (rhi .ge. 1.0d0 - 1.d-12)) then &&\
D3aMin = min(D3aLL, D3aL, D3a, D3aR) &&\
D3aMax = max(D3aLL, D3aL, D3a, D3aR) &&\
if (C3 * max(abs(D3aMin), abs(D3aMax)) .le. D3aMax-D3aMin) then &&\
if (abs(daplus) .le. abs(a(i)) .and. abs(daminus) .le. abs(a(i))) then &&\
if (daplus*daminus .lt. 0) then &&\
aplus(i) = a(i) + daplus * rhi &&\
aminus(i) = a(i) - daminus * rhi &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) - 2.0d0*(1.0d0-rhi)*daplus - rhi*daminus &&\
else if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) + 2.0d0*(1.0d0-rhi)*daminus + rhi*daplus &&\
endif &&\
else &&\
if (daplus*daminus .lt. 0) then &&\
aplus(i) = a(i) &&\
aminus(i) = a(i) &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) &&\
else if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) &&\
endif &&\
endif &&\
endif &&\
endif &&\
trivial_rp(i-1) = .false. &&\
trivial_rp(i) = .false. &&\
else &&\
if (abs(daplus) .le. abs(a(i)) .and. abs(daminus) .le. abs(a(i))) then &&\
if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) + 2.0d0*daminus &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) - 2.0d0*daplus &&\
end if &&\
else &&\
if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) &&\
end if &&\
endif &&\
trivial_rp(i-1) = .false. &&\
trivial_rp(i) = .false. &&\
endif
! We use (20) of Colella & Sekora 2008 to check wether we are at a local maximum.
! If so, we make use of (22) and (23) to preserve accuracy there. Otherwise we use
! enhanced monotocinity preservation (26)
! #define MON_WITH_LOCAL_EXTREMUM(aminus, a, aplus, C) \
! if ((aplus(i)-a(i))*(a(i)-aminus(i)) .le. 0 .or. (a(i-1)-a(i))*(a(i)-a(i+1)) .le. 0) then &&\
! D2a = - (12.0d0*a(i) - 6.0d0*(aminus(i)+aplus(i))) && \
! D2aC = a(i-1) - 2.0d0*a(i) + a(i+1) &&\
! D2aL = a(i-2) - 2.0d0*a(i-1) + a(i) &&\
! D2aR = a(i) - 2.0d0*a(i+1) + a(i+2) &&\
! if (D2a .ne. 0 .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aC) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aL) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aR)) then &&\
! aplus(i) = a(i) + (aplus(i) -a(i)) * sign(1.0d0, D2a) * min(C*abs(D2aL), C*abs(D2aR), C*abs(D2aC), abs(D2a)) / D2a &&\
! aminus(i) = a(i) + (aminus(i)-a(i)) * sign(1.0d0, D2a) * min(C*abs(D2aL), C*abs(D2aR), C*abs(D2aC), abs(D2a)) / D2a &&\
! else &&\
! aplus(i) = a(i) &&\
! aminus(i) = a(i) &&\
! end if &&\
! trivial_rp(i-1) = .false. &&\
! trivial_rp(i) = .false. &&\
! else &&\
! if (abs(aplus(i)-a(i)) .ge. 2.0d0*abs(aminus(i)-a(i))) then &&\
! aplus(i) = a(i) - 2.0d0*(aminus(i) - a(i)) &&\
! else if (abs(aminus(i)-a(i)) .ge. 2.0d0*abs(aplus(i)-a(i))) then &&\
! aminus(i) = a(i) - 2.0d0*(aplus(i) - a(i)) &&\
! end if &&\
! trivial_rp(i-1) = .false. &&\
! trivial_rp(i) = .false. &&\
! endif
!! Monotonicity of PPM 2011 (McCorquodale & Colella 2011), Sec. 2.4.1, Eq. 23-34.
!! This does not use the check for deviations from a cubic. Thus, it gets away with only 3 stencil points!
#define MON_WITH_LOCAL_EXTREMUM(aminus, a, aplus, C) \
daplus = aplus(i)-a(i) &&\
daminus = a(i)-aminus(i) &&\
if (daplus*daminus .le. 0 .or. (a(i-2)-a(i))*(a(i)-a(i+2)) .le. 0) then &&\
D2a = - (12.0d0*a(i) - 6.0d0*(aminus(i)+aplus(i))) && \
D2aC = a(i-1) - 2.0d0*a(i) + a(i+1) &&\
D2aL = a(i-2) - 2.0d0*a(i-1) + a(i) &&\
D2aR = a(i) - 2.0d0*a(i+1) + a(i+2) &&\
if (sign(1.0d0, D2a) .eq. sign(1.0d0, D2aC) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aL) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aR)) then &&\
D2aLim = sign(1.0d0, D2a) * min(C*abs(D2aL), C*abs(D2aR), C*abs(D2aC), abs(D2a)) &&\
else &&\
D2aLim = 0 &&\
end if &&\
if (abs(D2a) .le. 1.d-12*max(abs(a(i-2)), abs(a(i-1)), abs(a(i)), abs(a(i+1)), abs(a(i+2)))) then &&\
rhi = 0 &&\
else &&\
rhi = D2aLim / D2a &&\
endif &&\
if (.not. (rhi .ge. 1.0d0 - 1.d-12)) then &&\
if (abs(daplus) .le. abs(a(i)) .and. abs(daminus) .le. abs(a(i))) then &&\
if (daplus*daminus .lt. 0) then &&\
aplus(i) = a(i) + daplus * rhi &&\
aminus(i) = a(i) - daminus * rhi &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) - 2.0d0*(1.0d0-rhi)*daplus - rhi*daminus &&\
else if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) + 2.0d0*(1.0d0-rhi)*daminus + rhi*daplus &&\
endif &&\
else &&\
if (daplus*daminus .lt. 0) then &&\
aplus(i) = a(i) &&\
aminus(i) = a(i) &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) &&\
else if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) &&\
endif &&\
endif &&\
endif &&\
trivial_rp(i-1) = .false. &&\
trivial_rp(i) = .false. &&\
else &&\
if (abs(daplus) .le. abs(a(i)) .and. abs(daminus) .le. abs(a(i))) then &&\
if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) + 2.0d0*daminus &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) - 2.0d0*daplus &&\
end if &&\
else &&\
if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) &&\
end if &&\
endif &&\
trivial_rp(i-1) = .false. &&\
trivial_rp(i) = .false. &&\
endif
if (use_enhanced_ppm .eq. 1) then
!! Constrain parabolic profiles, PPM 2011/2008
if (PPM3) then
do i = 3, nx - 2
MON_WITH_LOCAL_EXTREMUM(rhominus,rho,rhoplus, enhanced_ppm_C2)
MON_WITH_LOCAL_EXTREMUM(velxminus,velx,velxplus, enhanced_ppm_C2)
MON_WITH_LOCAL_EXTREMUM(velyminus,vely,velyplus, enhanced_ppm_C2)
MON_WITH_LOCAL_EXTREMUM(velzminus,velz,velzplus, enhanced_ppm_C2)
end do
if (poly .eq. 0) then
do i = 3, nx - 2
MON_WITH_LOCAL_EXTREMUM(epsminus,eps,epsplus, enhanced_ppm_C2)
end do
end if
else
do i = 4, nx - 3
MON_WITH_LOCAL_EXTREMUM_STENCIL4(rhominus,rho,rhoplus, enhanced_ppm_C2, enhanced_ppm_C3)
MON_WITH_LOCAL_EXTREMUM_STENCIL4(velxminus,velx,velxplus, enhanced_ppm_C2, enhanced_ppm_C3)
MON_WITH_LOCAL_EXTREMUM_STENCIL4(velyminus,vely,velyplus, enhanced_ppm_C2, enhanced_ppm_C3)
MON_WITH_LOCAL_EXTREMUM_STENCIL4(velzminus,velz,velzplus, enhanced_ppm_C2, enhanced_ppm_C3)
end do
if (poly .eq. 0) then
do i = 4, nx - 3
MON_WITH_LOCAL_EXTREMUM_STENCIL4(epsminus,eps,epsplus, enhanced_ppm_C2, enhanced_ppm_C3)
end do
endif
end if
! Apply flattening after constraining the parabolic profiles
if (PPM3) then !!$ Implement C&W, page 197, but with a workaround which allows to use stencil=3.
do i = 3, nx - 2
flatten = tilde_flatten(i)
if (abs(1.d0 - flatten) > 0.d0) then
trivial_rp(i-1) = .false.
trivial_rp(i) = .false.
end if
rhoplus(i) = flatten * rhoplus(i) + (1.d0 - flatten) * rho(i)
rhominus(i) = flatten * rhominus(i) + (1.d0 - flatten) * rho(i)
velxplus(i) = flatten * velxplus(i) + (1.d0 - flatten) * velx(i)
velxminus(i) = flatten * velxminus(i) + (1.d0 - flatten) * velx(i)
velyplus(i) = flatten * velyplus(i) + (1.d0 - flatten) * vely(i)
velyminus(i) = flatten * velyminus(i) + (1.d0 - flatten) * vely(i)
velzplus(i) = flatten * velzplus(i) + (1.d0 - flatten) * velz(i)
velzminus(i) = flatten * velzminus(i) + (1.d0 - flatten) * velz(i)
if (poly .eq. 0) then
epsplus(i) = flatten * epsplus(i) + (1.d0 - flatten) * eps(i)
epsminus(i) = flatten * epsminus(i) + (1.d0 - flatten) * eps(i)
end if
end do
else !!$ Really implement C&W, page 197; which requires stencil 4.
do i = 4, nx - 3
s=sign(1.d0, -dpress(i))
flatten = max(tilde_flatten(i), tilde_flatten(i+s))
if (abs(1.d0 - flatten) > 0.d0) then
trivial_rp(i-1) = .false.
trivial_rp(i) = .false.
end if
rhoplus(i) = flatten * rhoplus(i) + (1.d0 - flatten) * rho(i)
rhominus(i) = flatten * rhominus(i) + (1.d0 - flatten) * rho(i)
velxplus(i) = flatten * velxplus(i) + (1.d0 - flatten) * velx(i)
velxminus(i) = flatten * velxminus(i) + (1.d0 - flatten) * velx(i)
velyplus(i) = flatten * velyplus(i) + (1.d0 - flatten) * vely(i)
velyminus(i) = flatten * velyminus(i) + (1.d0 - flatten) * vely(i)
velzplus(i) = flatten * velzplus(i) + (1.d0 - flatten) * velz(i)
velzminus(i) = flatten * velzminus(i) + (1.d0 - flatten) * velz(i)
if (poly .eq. 0) then
epsplus(i) = flatten * epsplus(i) + (1.d0 - flatten) * eps(i)
epsminus(i) = flatten * epsminus(i) + (1.d0 - flatten) * eps(i)
end if
end do
end if
endif
if (use_enhanced_ppm .eq. 0) then
!! Constrain parabolic profiles, PPM 1984
do i = GRHydro_stencil, nx - GRHydro_stencil + 1
! original Colella&Woodward monotonicity preservation
MON(rhominus,rho,rhoplus)
MON(velxminus,velx,velxplus)
MON(velyminus,vely,velyplus)
MON(velzminus,velz,velzplus)
end do
if (poly .eq. 0) then
do i = GRHydro_stencil, nx - GRHydro_stencil + 1
MON(epsminus,eps,epsplus)
end do
end if
endif
if (check_for_trivial_rp .eq. 0) then
trivial_rp = .false.
end if
!!$ excision
do i = 1, nx
if (GRHydro_enable_internal_excision /= 0 .and. &
(hydro_excision_mask(i) .ne. 0)) then
if (i .gt. 1) then
trivial_rp(i-1)=.true.
end if
trivial_rp(i)=.true.
else
!!$ Do not optimize cond away by combining the 'if's. Fortran does not
!!$ have to follow the order of sub-expressions given here and might
!!$ access outside the array range
cond = .false.
if (i .gt. 1 .and. GRHydro_enable_internal_excision /= 0) then
cond = hydro_excision_mask(i-1) .ne. 0
end if
if (cond) then
rhominus(i)=rho(i)
rhoplus(i)=rho(i)
velxminus(i)=velx(i)
velxplus(i)=velx(i)
velyminus(i)=vely(i)
velyplus(i)=vely(i)
velzminus(i)=velz(i)
velzplus(i)=velz(i)
rhominus(i-1)=rho(i)
rhoplus(i-1)=rho(i)
velxminus(i-1)=velx(i)
velxplus(i-1)=velx(i)
velyminus(i-1)=vely(i)
velyplus(i-1)=vely(i)
velzminus(i-1)=velz(i)
velzplus(i-1)=velz(i)
if (poly .eq. 0) then
epsminus(i)=eps(i)
epsplus(i)=eps(i)
epsminus(i-1)=eps(i)
epsplus(i-1)=eps(i)
end if
else
cond = .false.
if ((i.gt.2) .and. (i.lt.nx) .and. GRHydro_enable_internal_excision /= 0) then
cond = (ppm_mppm .eq. 0) .and. (hydro_excision_mask(i-2) .ne. 0)
end if
if (cond) then
call PPM_TVD(rho(i-1), rho(i), rho(i+1), rhominus(i), rhoplus(i))
call PPM_TVD(velx(i-1), velx(i), velx(i+1), velxminus(i), velxplus(i))
call PPM_TVD(vely(i-1), vely(i), vely(i+1), velyminus(i), velyplus(i))
call PPM_TVD(velz(i-1), velz(i), velz(i+1), velzminus(i), velzplus(i))
if (poly .eq. 0) then
call PPM_TVD(eps(i-1), eps(i), eps(i+1), epsminus(i), epsplus(i))
end if
end if
end if
cond = .false.
if (i .lt. nx .and. GRHydro_enable_internal_excision /= 0) then
cond = hydro_excision_mask(i+1) .ne. 0
end if
if (cond) then
rhominus(i)=rho(i)
rhoplus(i)=rho(i)
velxminus(i)=velx(i)
velxplus(i)=velx(i)
velyminus(i)=vely(i)
velyplus(i)=vely(i)
velzminus(i)=velz(i)
velzplus(i)=velz(i)
rhominus(i+1)=rho(i)
rhoplus(i+1)=rho(i)
velxminus(i+1)=velx(i)
velxplus(i+1)=velx(i)
velyminus(i+1)=vely(i)
velyplus(i+1)=vely(i)
velzminus(i+1)=velz(i)
velzplus(i+1)=velz(i)
if (poly .eq. 0) then
epsminus(i)=eps(i)
epsplus(i)=eps(i)
epsminus(i+1)=eps(i)
epsplus(i+1)=eps(i)
endif
else
cond = .false.
if ((i.lt.nx-1) .and. (i.gt.1) .and. GRHydro_enable_internal_excision /= 0) then
cond = (ppm_mppm .eq. 0) .and. (hydro_excision_mask(i+2) .ne. 0)
end if
if (cond) then
call PPM_TVD(rho(i-1), rho(i), rho(i+1), rhominus(i), rhoplus(i))
call PPM_TVD(velx(i-1), velx(i), velx(i+1), velxminus(i), velxplus(i))
call PPM_TVD(vely(i-1), vely(i), vely(i+1), velyminus(i), velyplus(i))
call PPM_TVD(velz(i-1), velz(i), velz(i+1), velzminus(i), velzplus(i))
if (poly .eq. 0) then
call PPM_TVD(eps(i-1), eps(i), eps(i+1), epsminus(i), epsplus(i))
end if
end if
end if
end if
end do
return
end subroutine SimplePPM_1d
subroutine SimplePPM_tracer_1d(nx,dx,rho,velx,vely,velz, &
tracer,tracerminus,tracerplus,press)
USE GRHydro_Scalars
implicit none
DECLARE_CCTK_PARAMETERS
CCTK_INT :: nx
CCTK_REAL :: dx
CCTK_REAL, dimension(nx) :: rho,velx,vely,velz
CCTK_REAL, dimension(nx,number_of_tracers) :: tracer,tracerminus,tracerplus
CCTK_REAL :: tracerflatomega
CCTK_INT :: i,s,itracer
CCTK_REAL, dimension(nx) :: press,dpress,tilde_flatten
CCTK_REAL, dimension(nx,number_of_tracers) :: dmtracer, dtracer, tracerflat!, d2tracer
CCTK_REAL :: dpress2,w,flatten,dvel
CCTK_REAL :: eta, etatilde
!!$ Average slopes delta_m(a). See (1.7) of Colella and Woodward, p.178
!!$ This is the expression for an even grid.
do i = 2, nx - 1
dpress(i) = press(i+1) - press(i-1)
end do
do itracer=1,number_of_tracers
do i = 2, nx - 1
dtracer(i,itracer) = 0.5d0 * (tracer(i+1,itracer) - tracer(i-1,itracer))
! d2tracer(i,itracer) = (tracer(i+1) - 2.d0 * tracer(i) + tracer(i-1))! / 6.d0 / dx / dx
! ! since we use d2tracer only for the condition d2tracer(i+1)*d2tracer(i-1)<0
! ! the denominator is not necessary
enddo
enddo
!!$ Steepened slope. See (1.8) of Colella and Woodward, p.178
do itracer=1,number_of_tracers
do i = 2, nx - 1
if( (tracer(i+1,itracer) - tracer(i,itracer)) * &
(tracer(i,itracer) - tracer(i-1,itracer)) > 0.0d0 ) then
dmtracer(i,itracer) = sign(1.0d0,dtracer(i,itracer)) * &
min(abs(dtracer(i,itracer)), 2.0d0 * &
abs(tracer(i,itracer) - tracer(i-1,itracer)), &
2.0d0 * abs(tracer(i+1,itracer) - tracer(i,itracer)))
else
dmtracer(i,itracer) = 0.0d0
endif
end do
enddo
!!$ Initial boundary states. See (1.9) of Colella and Woodward, p.178
do itracer=1,number_of_tracers
do i = 2, nx - 2
tracerplus(i,itracer) = 0.5d0 * (tracer(i,itracer) + tracer(i+1,itracer)) + &
(dmtracer(i,itracer) - dmtracer(i+1,itracer)) / 6.d0
tracerminus(i+1,itracer) = tracerplus(i,itracer)
enddo
enddo
!!$Discontinuity steepening. See (1.14-17) of C&W.
!!$This is the detect routine which mat be activated with the ppm_detect parameter
!!$Note that this part really also depends on the grid being even.
!!$Note also that we do not have access to the gas constant gamma.
!!$So this is just dropped from eq. (3.2) of C&W.
!!$We can get around this by just rescaling the constant k0 (ppm_k0 here).
!!! We might play around with this for the tracer. CURRENTLY TURNED OFF
#if 0
if (ppm_detect .eq. 1000) then
do itracer=1,number_of_tracers
do i = 3, nx - 2
if ( (dtracer(i+1,itracer)*dtracer(i-1,itracer) > 0.d0) & !make sure this is not an extremum
.and.(abs(tracer(i+1,itracer)-tracer(i-1,itracer)) - & !this is to prevent steepening
!of relatively small composition jumps
ppm_epsilon_shock * min(tracer(i+1,itracer), tracer(i-1,itracer)) > 0.d0 ) &
.and. & ! the actual criterion from Plewa & Mueller
((tracer(i+1,itracer)-tracer(i-1,itracer)) / &
(tracer(i+2,itracer)-tracer(i-2,itracer)) > ppm_omega1 ) ) then
etatilde = (tracer(i-2,itracer) - tracer(i+2,itracer) + &
4.d0 * dtracer(i,itracer)) / (dtracer(i,itracer) * 12.d0)
write(*,*) "Additional Steepening in Zone: ",i
else
etatilde = 0.d0
end if
eta = max(0.d0, min(1.d0, ppm_eta1 * (etatilde - ppm_eta2)))
if (ppm_k0 * abs(dtracer(i,itracer)) * min(press(i-1),press(i+1)) < &
abs(dpress(i)) * min(tracer(i-1,itracer), tracer(i+1,itracer))) then
eta = 0.d0
end if
tracerminus(i,itracer) = tracerminus(i,itracer) * (1.d0 - eta) + &
(tracer(i-1,itracer) + 0.5d0 * dmtracer(i-1,itracer)) * eta
tracerplus(i,itracer) = tracerplus(i,itracer) * (1.d0 - eta) + &
(tracer(i+1,itracer) - 0.5d0 * dmtracer(i+1,itracer)) * eta
end do
enddo
end if
#endif
!!$ Zone flattening. See appendix of C&W, p. 197-8.
do i = 3, nx - 2
dpress2 = press(i+2) - press(i-2)
dvel = velx(i+1) - velx(i-1)
if ( (abs(dpress(i)) > ppm_epsilon * min(press(i-1),press(i+1))) .and. &
(dvel < 0.d0) ) then
w = 1.d0
else
w = 0.d0
end if
if (abs(dpress2) < ppm_small) then
tilde_flatten(i) = 1.d0
else
tilde_flatten(i) = max(0.d0, 1.d0 - w * max(0.d0, ppm_omega2 * &
(dpress(i) / dpress2 - ppm_omega1)))
end if
end do
if (PPM3) then
do itracer=1,number_of_tracers
do i = 3, nx - 2
flatten = tilde_flatten(i)
tracerplus(i,itracer) = flatten * tracerplus(i,itracer) &
+ (1.d0 - flatten) * tracer(i,itracer)
tracerminus(i,itracer) = flatten * tracerminus(i,itracer) &
+ (1.d0 - flatten) * tracer(i,itracer)
end do
enddo
else !!$ Really implement C&W, page 197; which requires stencil 4.
do itracer=1,number_of_tracers
do i = 4, nx - 3
s=sign(1.d0, -dpress(i))
flatten = max(tilde_flatten(i), tilde_flatten(i+s))
tracerplus(i,itracer) = flatten * tracerplus(i,itracer) + &
(1.d0 - flatten) * tracer(i,itracer)
tracerminus(i,itracer) = flatten * tracerminus(i,itracer) &
+ (1.d0 - flatten) * tracer(i,itracer)
end do
enddo
end if
!! Additional flattening a la Plewa & Mueller
#if 1
do itracer=1,number_of_tracers
do i = 2, nx - 1
if ( ( tracer(i+1,itracer) - tracer(i,itracer) ) * &
( tracer(i,itracer) - tracer(i-1,itracer) ) < 0.0d0 ) then
tracerflat(i,itracer) = 1.0d0
else
tracerflat(i,itracer) = 0.0d0
endif
enddo
enddo
do itracer=1,number_of_tracers
do i = 3, nx -2
tracerflatomega = 0.5d0 * max(tracerflat(i-1,itracer),2.0d0*tracerflat(i,itracer), &
tracerflat(i+1,itracer)) * ppm_omega_tracer
tracerplus(i,itracer) = tracerflatomega*tracer(i,itracer) + &
(1.0d0 - tracerflatomega)*tracerplus(i,itracer)
tracerminus(i,itracer) = tracerflatomega*tracer(i,itracer) + &
(1.0d0 - tracerflatomega)*tracerminus(i,itracer)
enddo
enddo
#endif
!!$ Monotonicity. See (1.10) of C&W.
do itracer=1,number_of_tracers
do i = GRHydro_stencil, nx - GRHydro_stencil + 1
if (((tracerplus(i,itracer)-tracer(i,itracer))* &
(tracer(i,itracer)-tracerminus(i,itracer)) .le. 0.d0)) then
tracerminus(i,itracer) = tracer(i,itracer)
tracerplus(i,itracer) = tracer(i,itracer)
else if ((tracerplus(i,itracer) - tracerminus(i,itracer)) * (tracer(i,itracer) - 0.5d0 * &
(tracerplus(i,itracer) + tracerminus(i,itracer))) > &
(tracerplus(i,itracer) - tracerminus(i,itracer))**2 / 6.d0) then
tracerminus(i,itracer) = 3.d0 * tracer(i,itracer) - 2.d0 * tracerplus(i,itracer)
else if ((tracerplus(i,itracer) - tracerminus(i,itracer)) * (tracer(i,itracer) - 0.5d0 * &
(tracerplus(i,itracer) + tracerminus(i,itracer))) < &
-(tracerplus(i,itracer) - tracerminus(i,itracer))**2 / 6.d0 ) then
tracerplus(i,itracer) = 3.d0 * tracer(i,itracer) - 2.d0 * tracerminus(i,itracer)
end if
end do
enddo
end subroutine SimplePPM_tracer_1d
subroutine SimplePPM_ye_1d(nx,dx,rho,velx,vely,velz, &
Y_e,Y_e_minus,Y_e_plus,press)
USE GRHydro_Scalars
implicit none
DECLARE_CCTK_PARAMETERS
CCTK_INT :: nx
CCTK_REAL :: dx
CCTK_REAL, dimension(nx) :: rho,velx,vely,velz
CCTK_REAL, dimension(nx) :: Y_e,Y_e_minus,Y_e_plus
CCTK_REAL :: Y_eflatomega
CCTK_INT :: i,s
CCTK_REAL, dimension(nx) :: press,dpress,tilde_flatten
CCTK_REAL, dimension(nx) :: dmY_e, dY_e, Y_eflat!, d2tracer
CCTK_REAL :: dpress2,w,flatten,dvel
CCTK_REAL :: eta, etatilde
CCTK_REAL :: D2a, D2aL, D2aR, D2aC, D2aLim, rhi, threshold, rho_min, daplus, daminus, D3a, D3aLL, D3aL, D3aR, D2aLL, D2aRR, D3aMin, D3aMax
do i = 2, nx - 1
dpress(i) = press(i+1) - press(i-1)
end do
if (use_enhanced_PPM .eq. 0) then
!!$ Average slopes delta_m(a). See (1.7) of Colella and Woodward, p.178
!!$ This is the expression for an even grid.
do i = 2, nx - 1
dY_e(i) = 0.5d0 * (Y_e(i+1) - Y_e(i-1))
! d2Y_e(i,iY_e) = (Y_e(i+1) - 2.d0 * Y_e(i) + Y_e(i-1))! / 6.d0 / dx / dx
! ! since we use d2Y_e only for the condition d2Y_e(i+1)*d2Y_e(i-1)<0
! ! the denominator is not necessary
enddo
!!$ Steepened slope. See (1.8) of Colella and Woodward, p.178
do i = 2, nx - 1
if( (Y_e(i+1) - Y_e(i)) * &
(Y_e(i) - Y_e(i-1)) > 0.0d0 ) then
dmY_e(i) = sign(1.0d0,dY_e(i)) * &
min(abs(dY_e(i)), 2.0d0 * &
abs(Y_e(i) - Y_e(i-1)), &
2.0d0 * abs(Y_e(i+1) - Y_e(i)))
else
dmY_e(i) = 0.0d0
endif
end do
!!$ Initial boundary states. See (1.9) of Colella and Woodward, p.178
do i = 2, nx - 2
Y_e_plus(i) = 0.5d0 * (Y_e(i) + Y_e(i+1)) + &
(dmY_e(i) - dmY_e(i+1)) / 6.d0
Y_e_minus(i+1) = Y_e_plus(i)
enddo
else
!! This is the modified PPM algorithm by Colella & Sekora 2008.
!! This uses a better limiter based on second derivatives that preserves
!! accuracy at local extrema. It also uses a higher-order interpolation polynomial.
!!$ Initial boundary states (sixth order accurate). See (17) of Colella and Sekora, p.7071
#undef APPROX_AT_CELL_INTERFACE_STENCIL4
!!$ Initial boundary states (sixth order accurate). See (17) of Colella and Sekora 2008, p.7071
#define APPROX_AT_CELL_INTERFACE_STENCIL4(a, ah) \
ah = 37.0d0/60.0d0*(a(i)+a(i+1)) - 2.0d0/15.0d0*(a(i-1)+a(i+2)) + 1.0d0/60.0d0*(a(i-2)+a(i+3))
#undef APPROX_AT_CELL_INTERFACE
!!$ Initial boundary states (4th order accurate). See (16) of Colella and Sekora 2008, p.7071
#define APPROX_AT_CELL_INTERFACE(a, ah) \
ah = 7.0d0/12.0d0*(a(i)+a(i+1)) - 1.0d0/12.0d0*(a(i-1)+a(i+2))
#undef LIMIT
#define LIMIT(a,ah,C, alim) \
if ((min(a(i),a(i+1)) .le. ah) .and. (ah .le. max(a(i),a(i+1)))) then &&\
alim = ah &&\
else &&\
D2a = 3.0d0 * (a(i) - 2.0d0*ah + a(i+1)) &&\
D2aL = (a(i-1) - 2.0d0*a(i) + a(i+1)) &&\
D2aR = (a(i) - 2.0d0*a(i+1) + a(i+2)) &&\
if (D2a*D2aR .ge. 0 .and. D2a*D2aL .ge. 0 .and. abs(D2aLim) .lt. abs(0.5d0*(a(i)+a(i+1)))) then &&\
alim = 0.5d0*(a(i)+a(i+1)) - sign(1.0d0, D2a)*min(C*abs(D2aL), C*abs(D2aR), abs(D2a))/3.0d0 &&\
else &&\
alim = 0.5d0*(a(i)+a(i+1)) &&\
end if &&\
endif
if (PPM3) then
!! We initialize "plus" \equiv a_j+1/2 with (16) via APPROX_AT_CELL_INTERFACE,
!! then checking for (13) of Colella & Sekora 2008 and applying
!! (18) and (19) if (13) is not satisfied. This is done with LIMIT.
!! There, we also switch to lower order if we are near atmosphere values.
do i = 2, nx-2
APPROX_AT_CELL_INTERFACE(Y_e, Y_e_plus(i))
LIMIT(Y_e, Y_e_plus(i), enhanced_ppm_C2, Y_e_plus(i))
Y_e_minus(i+1) = Y_e_plus(i)
end do
else
!! Same as above but for 4 stencil points using (17) of Colella & Sekora 2008 as
!! initial states.
do i = 3, nx-3
APPROX_AT_CELL_INTERFACE_STENCIL4(Y_e, Y_e_plus(i))
LIMIT(Y_e, Y_e_plus(i), enhanced_ppm_C2, Y_e_plus(i))
Y_e_minus(i+1) = Y_e_plus(i)
end do
endif
end if
!!$Discontinuity steepening. See (1.14-17) of C&W.
!!$This is the detect routine which mat be activated with the ppm_detect parameter
!!$Note that this part really also depends on the grid being even.
!!$Note also that we do not have access to the gas constant gamma.
!!$So this is just dropped from eq. (3.2) of C&W.
!!$We can get around this by just rescaling the constant k0 (ppm_k0 here).
!!$ Zone flattening. See appendix of C&W, p. 197-8.
do i = 3, nx - 2
dpress2 = press(i+2) - press(i-2)
dvel = velx(i+1) - velx(i-1)
if ( (abs(dpress(i)) > ppm_epsilon * min(press(i-1),press(i+1))) .and. &
(dvel < 0.d0) ) then
w = 1.d0
else
w = 0.d0
end if
if (abs(dpress2) < ppm_small) then
tilde_flatten(i) = 1.d0
else
tilde_flatten(i) = max(0.d0, 1.d0 - w * max(0.d0, ppm_omega2 * &
(dpress(i) / dpress2 - ppm_omega1)))
end if
end do
if (use_enhanced_ppm .eq. 0) then
if (PPM3) then
do i = 3, nx - 2
flatten = tilde_flatten(i)
Y_e_plus(i) = flatten * Y_e_plus(i) &
+ (1.d0 - flatten) * Y_e(i)
Y_e_minus(i) = flatten * Y_e_minus(i) &
+ (1.d0 - flatten) * Y_e(i)
end do
else !!$ Really implement C&W, page 197; which requires stencil 4.
do i = 4, nx - 3
s=sign(1.d0, -dpress(i))
flatten = max(tilde_flatten(i), tilde_flatten(i+s))
Y_e_plus(i) = flatten * Y_e_plus(i) + &
(1.d0 - flatten) * Y_e(i)
Y_e_minus(i) = flatten * Y_e_minus(i) &
+ (1.d0 - flatten) * Y_e(i)
end do
end if
endif
#undef MON_WITH_LOCAL_EXTREMUM_STENCIL4
!! Monotonicity of PPM 2011 (McCorquodale & Colella 2011), Sec. 2.4.1, Eq. 23-34
!! This requires 4 stencil points.
#define MON_WITH_LOCAL_EXTREMUM_STENCIL4(aminus, a, aplus, C, C3) \
daplus = aplus(i)-a(i) &&\
daminus = a(i)-aminus(i) &&\
if (daplus*daminus .le. 0 .or. (a(i-2)-a(i))*(a(i)-a(i+2)) .le. 0) then &&\
D2a = - (12.0d0*a(i) - 6.0d0*(aminus(i)+aplus(i))) && \
D2aC = a(i-1) - 2.0d0*a(i) + a(i+1) &&\
D2aL = a(i-2) - 2.0d0*a(i-1) + a(i) &&\
D2aLL = a(i-3) - 2.0d0*a(i-2) + a(i-1) &&\
D2aR = a(i) - 2.0d0*a(i+1) + a(i+2) &&\
D2aRR = a(i+1) - 2.0d0*a(i+2) + a(i+3) &&\
D3a = D2aR - D2aC &&\
D3aL = D2aC - D2aL &&\
D3aR = D2aRR - D2aR &&\
D3aLL = D2aL - D2aLL &&\
if (sign(1.0d0, D2a) .eq. sign(1.0d0, D2aC) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aL) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aR)) then &&\
D2aLim = sign(1.0d0, D2a) * min(C*abs(D2aL), C*abs(D2aR), C*abs(D2aC), abs(D2a)) &&\
else &&\
D2aLim = 0 &&\
end if &&\
if (abs(D2a) .le. 1.d-12*max(abs(a(i-2)), abs(a(i-1)), abs(a(i)), abs(a(i+1)), abs(a(i+2)))) then &&\
rhi = 0 &&\
else &&\
rhi = D2aLim / D2a &&\
endif &&\
if (.not. (rhi .ge. 1.0d0 - 1.d-12)) then &&\
D3aMin = min(D3aLL, D3aL, D3a, D3aR) &&\
D3aMax = max(D3aLL, D3aL, D3a, D3aR) &&\
if (C3 * max(abs(D3aMin), abs(D3aMax)) .le. D3aMax-D3aMin) then &&\
if (abs(daplus) .le. abs(a(i)) .and. abs(daminus) .le. abs(a(i))) then &&\
if (daplus*daminus .lt. 0) then &&\
aplus(i) = a(i) + daplus * rhi &&\
aminus(i) = a(i) - daminus * rhi &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) - 2.0d0*(1.0d0-rhi)*daplus - rhi*daminus &&\
else if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) + 2.0d0*(1.0d0-rhi)*daminus + rhi*daplus &&\
endif &&\
else &&\
if (daplus*daminus .lt. 0) then &&\
aplus(i) = a(i) &&\
aminus(i) = a(i) &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) &&\
else if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) &&\
endif &&\
endif &&\
endif &&\
endif &&\
else &&\
if (abs(daplus) .le. abs(a(i)) .and. abs(daminus) .le. abs(a(i))) then &&\
if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) + 2.0d0*daminus &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) - 2.0d0*daplus &&\
end if &&\
else &&\
if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) &&\
end if &&\
endif &&\
endif
#undef MON_WITH_LOCAL_EXTREMUM
!! Monotonicity of PPM 2011 (McCorquodale & Colella 2011), Sec. 2.4.1, Eq. 23-34.
!! This does not use the check for deviations from a cubic. Thus, it gets away with only 3 stencil points!
#define MON_WITH_LOCAL_EXTREMUM(aminus, a, aplus, C) \
daplus = aplus(i)-a(i) &&\
daminus = a(i)-aminus(i) &&\
if (daplus*daminus .le. 0 .or. (a(i-2)-a(i))*(a(i)-a(i+2)) .le. 0) then &&\
D2a = - (12.0d0*a(i) - 6.0d0*(aminus(i)+aplus(i))) && \
D2aC = a(i-1) - 2.0d0*a(i) + a(i+1) &&\
D2aL = a(i-2) - 2.0d0*a(i-1) + a(i) &&\
D2aR = a(i) - 2.0d0*a(i+1) + a(i+2) &&\
if (sign(1.0d0, D2a) .eq. sign(1.0d0, D2aC) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aL) .and. sign(1.0d0, D2a) .eq. sign(1.0d0, D2aR)) then &&\
D2aLim = sign(1.0d0, D2a) * min(C*abs(D2aL), C*abs(D2aR), C*abs(D2aC), abs(D2a)) &&\
else &&\
D2aLim = 0 &&\
end if &&\
if (abs(D2a) .le. 1.d-12*max(abs(a(i-2)), abs(a(i-1)), abs(a(i)), abs(a(i+1)), abs(a(i+2)))) then &&\
rhi = 0 &&\
else &&\
rhi = D2aLim / D2a &&\
endif &&\
if (.not. (rhi .ge. 1.0d0 - 1.d-12)) then &&\
if (abs(daplus) .le. abs(a(i)) .and. abs(daminus) .le. abs(a(i))) then &&\
if (daplus*daminus .lt. 0) then &&\
aplus(i) = a(i) + daplus * rhi &&\
aminus(i) = a(i) - daminus * rhi &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) - 2.0d0*(1.0d0-rhi)*daplus - rhi*daminus &&\
else if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) + 2.0d0*(1.0d0-rhi)*daminus + rhi*daplus &&\
endif &&\
else &&\
if (daplus*daminus .lt. 0) then &&\
aplus(i) = a(i) &&\
aminus(i) = a(i) &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) &&\
else if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) &&\
endif &&\
endif &&\
endif &&\
else &&\
if (abs(daplus) .le. abs(a(i)) .and. abs(daminus) .le. abs(a(i))) then &&\
if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) - 2.0d0*(aminus(i) - a(i)) &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) - 2.0d0*(aplus(i) - a(i)) &&\
end if &&\
else &&\
if (abs(daplus) .ge. 2.0d0*abs(daminus)) then &&\
aplus(i) = a(i) &&\
else if (abs(daminus) .ge. 2.0d0*abs(daplus)) then &&\
aminus(i) = a(i) &&\
end if &&\
endif &&\
endif
if (use_enhanced_ppm .eq. 1) then
! Constrain parabolic profiles
if (PPM3) then
do i = GRHydro_stencil, nx - GRHydro_stencil + 1
MON_WITH_LOCAL_EXTREMUM(Y_e_minus,Y_e,Y_e_plus, enhanced_ppm_C2)
end do
else
do i = GRHydro_stencil, nx - GRHydro_stencil + 1
MON_WITH_LOCAL_EXTREMUM_STENCIL4(Y_e_minus,Y_e,Y_e_plus, enhanced_ppm_C2, enhanced_ppm_C3)
end do
endif
! In PPM 2011/2008, flattening is applied after constraining parabolic profiles.
if (PPM3) then
do i = 3, nx - 2
flatten = tilde_flatten(i)
Y_e_plus(i) = flatten * Y_e_plus(i) &
+ (1.d0 - flatten) * Y_e(i)
Y_e_minus(i) = flatten * Y_e_minus(i) &
+ (1.d0 - flatten) * Y_e(i)
end do
else !!$ Really implement C&W, page 197; which requires stencil 4.
do i = 4, nx - 3
s=sign(1.d0, -dpress(i))
flatten = max(tilde_flatten(i), tilde_flatten(i+s))
Y_e_plus(i) = flatten * Y_e_plus(i) + &
(1.d0 - flatten) * Y_e(i)
Y_e_minus(i) = flatten * Y_e_minus(i) &
+ (1.d0 - flatten) * Y_e(i)
end do
end if
else
!!$ Monotonicity. See (1.10) of C&W.
do i = GRHydro_stencil, nx - GRHydro_stencil + 1
if (((Y_e_plus(i)-Y_e(i))* &
(Y_e(i)-Y_e_minus(i)) .le. 0.d0)) then
Y_e_minus(i) = Y_e(i)
Y_e_plus(i) = Y_e(i)
else if ((Y_e_plus(i) - Y_e_minus(i)) * (Y_e(i) - 0.5d0 * &
(Y_e_plus(i) + Y_e_minus(i))) > &
(Y_e_plus(i) - Y_e_minus(i))**2 / 6.d0) then
Y_e_minus(i) = 3.d0 * Y_e(i) - 2.0d0 * Y_e_plus(i)
else if ((Y_e_plus(i) - Y_e_minus(i)) * (Y_e(i) - 0.5d0 * &
(Y_e_plus(i) + Y_e_minus(i))) < &
-(Y_e_plus(i) - Y_e_minus(i))**2 / 6.0d0 ) then
Y_e_plus(i) = 3.d0 * Y_e(i) - 2.d0 * Y_e_minus(i)
end if
end do
end if
end subroutine SimplePPM_ye_1d
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