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flux.f90
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module flow
use numerics
implicit none
contains
function finit(x)
real(rp) , intent(in):: x
real(rp) :: finit
if ( x < 0._rp ) then
finit=un
else
finit=-un
end if
return
end function finit
function finit_bis(x)
real(rp) , intent(in):: x
real(rp) :: finit_bis
if ( x <=0._rp ) then
finit_bis=0.5_rp
else
finit_bis=zero
end if
return
end function finit_bis
subroutine finit_save()
!---------------INITIALISATION DONNEE INITIALE-------------------------
integer :: i
do i=1, N
u_prev(i)=finit(x(i))
end do
!-------------------------STOCKAGE U0----------------------------------
call save_file(x,u_prev,'u_init.txt' )
end subroutine finit_save
function f(u)
real(rp) , intent(in):: u
real(rp) :: f
f=u**3 - u
return
end function f
function fp(x)
real(rp) , intent(in):: x
real(rp) :: fp
fp=3._rp*x**2 - un
return
end function fp
function minmod(ug,uc,ud,dx)
real(rp) , intent(in):: ug,uc,ud, dx
real(rp) :: init, sr, sl, minmod
sr=(uc-ud)/dx
sl=(ug-uc)/dx
if( sl>zero .and. sr >zero)then
minmod=min(sl,sr)
else if( sl<zero .and. sr <zero) then
minmod=max(sl,sr)
else
minmod=zero
end if
return
end function minmod
subroutine LF(ul,ur,dx,dt,g) ! Flux LAX FRIEDRIECH
real(rp), intent(in) :: ul, ur, dx , dt
real(rp), intent(out) :: g
g=0.5*(f(ul)+f(ur))-0.5*(dx/dt)*(ur-ul) ! Lax-Friedriech flow
return
end subroutine LF
subroutine Godu(ul,ur,i, g) !! Flux Godunov
real(rp), intent(in) :: ul, ur
integer , intent(in) :: i
real(rp), intent(out) :: g
real(rp) :: sigma, u_star
!-------------------------f convexe-------------------------------------
if(ul== ur)then
g=f(ul)
return
end if
if(ul > zero .and. ur > zero ) then
if(ul < ur)then
if(i*dx < t*fp(ul)) then
g=f(ul)
else if( t*fp(ul)< i*dx .and. i*dx< t*fp(ur) ) then
g=sqrt(((i*dx/t)+un)/3._rp)
else if ( t*fp(ur) < i*dx )then
g=f(ur)
end if
else if (ur < ul)then
sigma=ur**2+ur*ul+ul**2 -un
if(i*dx< t*sigma)then
g=f(ul)
else if(t*sigma < i*dx)then
g=f(ur)
end if
end if
!-----------------------------f concave---------------------------------
else if (ul < zero .and. ur < zero ) then
if(ul > ur)then
if(i*dx< t*fp(ul))then !detente
g=f(ul)
else if( t*fp(ul)< i*dx .and. i*dx < t*fp(ur) )then
g=f(sqrt(((i*dx/t)+un)/3._rp))
else
g=f(ur)
end if
else if (ul < ur)then ! choc
sigma=ur**2+ur*ul+ul**2 -un
if(i*dx< t*sigma)then
g=f(ul)
else if (t*sigma < i*dx) then
g=f(ur)
end if
end if
!----------------------f ni concave ni convexe--------------------------
else if (ul < zero .and. ur > zero ) then
u_star=-0.5_rp*ur
sigma=(f(u_star)-f(ul))/(u_star-ul)
if( i*dx < fp(ul)*t )then
g=f(ul)
else if( fp(ul)*t < i*dx .and. i*dx < t*sigma )then
g=f(u_star)
else if ( sigma*t < i*dx .and. i*dx < fp(ur)*t ) then
g=f(sqrt(((i*dx/t)+un)/3._rp))
else if ( fp(ur)*t < i*dx) then
g=f(ur)
end if
else if( zero < ul .and. ur < zero ) then
u_star=-0.5_rp*ur
sigma=(f(u_star)-f(ul))/(u_star-ul)
if(i*dx< t*f(ul) )then
g=f(ul)
else if(t*f(ul)< i*dx .and. i*dx < t*sigma) then
g=f(u_star)
else if( t*sigma< i*dx .and. i*dx<t*fp(ur) )then
g=f(sqrt(((i*dx/t)+un)/3._rp))
else if( t*fp(ur)< i*dx )then
g=f(ur)
end if
end if
end subroutine Godu
SUBROUTINE LF_solve()
INTEGER :: i
t=zero
DO i=1,N
u_prev(i)=finit(g+i*dx)
END DO
DO WHILE (t<Tmax)
!---------------Dt EN FONCTION DE LA CFL-----------------------------
DO i=1,N
tmp(i)=fp(u_prev(i))
END DO
dt=0.5_rp*dx/abs(maxval(tmp))
!--------------------VECTEUR DE FLUX------------------------------------
flux(1)=f(u_prev(1)) ! BORD GAUCHE
DO i=2, N
CALL LF(u_prev(i-1),u_prev(i),dx,dt,flux(i)) ! FLUX NUMERIQUE
END DO
!flux(N)=f(u_prev(N)) ! BORD DROIT
!---------------------MISE A JOUR---------------------------------------
DO i=1, N-1
u_next(i)=u_prev(i)-(dt/dx)*(flux(i+1)-flux(i))
END DO
u_next(N)=u_prev(N)-(dt/dx)*(flux(N)-flux(N-1))
u_prev=u_next
t=t+dt
END DO
!-------------------------Sauvegarde ----------------------------------
CALL save_file(x,u_next, 'u_approx_LF.txt')
END SUBROUTINE LF_solve
SUBROUTINE Godu_solve()
INTEGER :: k
t=zero
DO k=1,N
u_prev(k)=0.5_rp*finit(g+k*dx)
END DO
DO WHILE (t<Tmax)
!write(*,*) t
!---------------Dt EN FONCTION DE LA CFL-----------------------------
DO k=1,N
tmp(k)=abs(fp(u_prev(k)) )
END DO
dt=0.5_rp*dx/maxval(tmp)
!--------------------VECTEUR DE FLUX------------------------------------
! BORD GAUCHE
flux(0)=f(u_prev(1))
CALL Godu(u_prev(1),u_prev(2),k,flux(1))
DO k=2, N-1
CALL Godu(u_prev(k-1),u_prev(k),k,flux(k)) ! FLUX NUMERIQUE
END DO
CALL Godu(u_prev(N-1),u_prev(N),N,flux(N)) ! BORD DROIT
!---------------------MISE A JOUR---------------------------------------
!write(*,*) maxval(flux)
u_next(1)=u_prev(1)-(dt/dx)*(flux(1)-flux(0))
DO k=2, N
u_next(k)=u_prev(k)-(dt/dx)*(flux(k)-flux(k-1))
END DO
u_prev=u_next
t=t+dt
END DO
!-------------------------Sauvegarde ----------------------------------
CALL save_file(x,u_next, 'u_approx_Godunov.txt')
END SUBROUTINE Godu_solve
subroutine Muscl()
real(rp), dimension(:), allocatable :: u, uplus, umoin, tmp
real(rp) :: sigma, gg , dd
integer :: i
allocate(uplus(n))
allocate(umoin(n))
allocate(u(n))
allocate(tmp(n))
DO i=1,N
u(i)=finit(g+i*dx)
END DO
Do While(t<Tmax)
!---------------Dt EN FONCTION DE LA CFL-----------------------------
DO i=1,N
tmp(i)=fp(u(i))
END DO
dt=0.5_rp*dx/abs(maxval(tmp))
!-----------------------------------------------------------------------
umoin(1)=u(1)-0.5_rp*dx*fp(u(1))
uplus(1)=u(1)+0.5_rp*dx*fp(u(1))
do i=2,n-1
sigma=minmod(u(i-1),u(i),u(i+1),dx)
umoin(i)=u(i)-0.5_rp*dx*sigma
uplus(i)=u(i)+0.5_rp*dx*sigma
end do
umoin(n)=u(n)-0.5_rp*dx*sigma
uplus(n)=u(n)+0.5_rp*dx*sigma
!--------------BOUCLE EN ESPACE----------------------------------------
do i=2,n-1
CALL LF(uplus(i),umoin(i+1),dx,dt,gg)
CALL LF(uplus(i-1),umoin(i),dx,dt,dd)
u(i)=u(i)-(dt/dx)*(gg-dd)
end do
CALL LF(uplus(n),umoin(n),dx,dt,gg)
CALL LF(uplus(n-1),umoin(n),dx,dt,dd)
u(n)=u(n)-(dt/dx)*(gg-dd)
t=t+dt
END DO
Call save_file(x,u, 'Muscl.txt')
end subroutine
end module flow