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CUEuler.f90
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! Author of the program Vladimir Fuka
! Modified by
! Donato Cecere
! 1 July 2007
module CUeuler
implicit none
integer,parameter:: KND=KIND(1.0D0)
integer RKtype !(1 - 1. order ... 3 - 3. order, 4. fully discrete)
integer FLUXtype !(1 - KURGTADMOR, 2 - CENTUPW, 3 - LAXFRIED, 4 - LAXWENDROFF,5 - MUSTA)
integer MUSTAstages!(Number of stages for MUSTA flux, for 1 it reduces to FORCE flux)
integer RECtype !(1 - piecewise constant, 2 - piecewise linear)
integer limitertype !(1 - minmod, 2 - modified minmod (MC), 3 - modified superbee)
real(KND) limparam !(parameter for limiter - only if needed)
integer extype !(1 - shock tube test, 2 - smooth test case with periodic BC)
integer outputtype !(1 - single frame textfiles for gnuplot, 2- one Tecplot file) Tecplot output by Donato Cecere, beta
integer nx,nt
real(KND) time,dx,xl,xr,dt,tend,CFL,rhoLEFT,rhouLEFT,eLEFT,rhoRIGHT,rhouRIGHT,eRIGHT
real(KND),parameter:: gamma=1.4_KND, pi=3.1415926535_KND
interface MINMOD
module procedure MINMOD2, MINMOD3
end interface
contains
subroutine TMARCH(rho,rhou,e,x)
integer i
real(KND),dimension(-2:):: rho,rhou,e,x
!real(KND),dimension(-2:nx+3):: rho,rhou,e
real(KND),dimension(LBOUND(rho,1):UBOUND(rho,1)):: rho2,rhou2,e2,drho,drhou,de
if (RKtype==1) then
call DRK(drho,drhou,de,rho,rhou,e,x)
rho=rho + dt*drho
rhou=rhou + dt*drhou
e=e + dt*de
elseif (RKtype==2) then
call DRK(drho,drhou,de,rho,rhou,e,x)
rho2=rho + dt*drho
rhou2=rhou + dt*drhou
e2=e + dt*de
call DRK(drho,drhou,de,rho2,rhou2,e2,x)
rho=(1._KND/2._KND)*rho + (1._KND/2._KND)*rho2 + (1._KND/2._KND)*dt*drho
rhou=(1._KND/2._KND)*rhou + (1._KND/2._KND)*rhou2 + (1._KND/2._KND)*dt*drhou
e=(1._KND/2._KND)*e + (1._KND/2._KND)*e2 + (1._KND/2._KND)*dt*de
elseif (RKtype==3) then
call DRK(drho,drhou,de,rho,rhou,e,x) !1
rho2=rho + dt*drho
rhou2=rhou + dt*drhou
e2=e + dt*de
call DRK(drho,drhou,de,rho2,rhou2,e2,x) !2
rho2=(3._KND/4._KND)*rho + (1._KND/4._KND)*rho2 + (1._KND/4._KND)*dt*drho
rhou2=(3._KND/4._KND)*rhou + (1._KND/4._KND)*rhou2 + (1._KND/4._KND)*dt*drhou
e2=(3._KND/4._KND)*e + (1._KND/4._KND)*e2 + (1._KND/4._KND)*dt*de
call DRK(drho,drhou,de,rho2,rhou2,e2,x) !3
rho=(1._KND/3._KND)*rho + (2._KND/3._KND)*rho2 + (2._KND/3._KND)*dt*drho
rhou=(1._KND/3._KND)*rhou + (2._KND/3._KND)*rhou2 + (2._KND/3._KND)*dt*drhou
e=(1._KND/3._KND)*e + (2._KND/3._KND)*e2 + (2._KND/3._KND)*dt*de
elseif (RKtype==4) then
call DFD(rho,rhou,e)
endif
endsubroutine TMARCH
subroutine DRK(drho,drhou,de,rho,rhou,e,x)
real(KND),dimension(-2:):: rho,rhou,e,drho,drhou,de,x
real(KND),allocatable,dimension(:):: rhoL,rhouL,eL,rhoR,rhouR,eR
drho=0
drhou=0
de=0
allocate(rhoL(-2:nx+2))
allocate(rhouL(-2:nx+2))
allocate(eL(-2:nx+2))
allocate(rhoR(-2:nx+2))
allocate(rhouR(-2:nx+2))
allocate(eR(-2:nx+2))
call BOUND(rho,rhou,e) !boundary condition
if (RECtype==1) then
call PCONST(rhoL,rhouL,eL,rhoR,rhouR,eR,rho,rhou,e)
elseif (RECtype==2) then
call MUSCL(rhoL,rhouL,eL,rhoR,rhouR,eR,rho,rhou,e,x)
endif
if (FLUXtype==1) then
call FLUXESKT(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR,x)
elseif (FLUXtype==2) then
call FLUXESCU(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR)
elseif (FLUXtype==3) then
call FLUXESLF(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR)
elseif (FLUXtype==4) then
call FLUXESLW(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR)
elseif (FLUXtype==5) then
call FLUXESMUSTA(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR)
elseif (FLUXtype==6) then
call FLUXESCULD(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR)
endif
deallocate(rhoL,rhoR,rhouL,rhouR,eL,eR)
endsubroutine DRK
subroutine DFD(rho,rhou,e)
real(KND),dimension(-2:):: rho,rhou,e
if (FLUXtype==3) then
call FDLF(rho,rhou,e)
elseif (FLUXtype==4) then
call FDLW(rho,rhou,e)
endif
endsubroutine DFD
! This procedure calculates extrapolated values of the dependent variables using
! a central difference scheme: k = 1, see Tannehill-Anderson-Pletcher, pp. 204-205.
! The values for the extrapolated variables on both sides of the interface are
! calculated on the left of i point.
subroutine MUSCL(rhoL,rhouL,eL,rhoR,rhouR,eR,rho,rhou,e,x)
real(KND),dimension(-2:)::rhoL,rhouL,eL,rhoR,rhouR,eR,rho,rhou,e,x
real(KND) sL,sC,sR
integer i
do i=0,nx+1
! sL=(rho(i)-rho(i-1))
sL=(rho(i)-rho(i-1))/(x(i)-x(i-1))
! sR=(rho(i+1)-rho(i))
sR=(rho(i+1)-rho(i))/(x(i+1)-x(i))
sL=sL
sR=sR
sC=LIMITER(sL,sR)
rhoR(i)=rho(i)-sC*(0.5_KND)*(x(i)-x(i-1))
rhoL(i+1)=rho(i)+sC*(0.5_KND)*(x(i+1)-x(i))
enddo
do i=0,nx+1
! sL=(rhou(i)-rhou(i-1))
sL=(rhou(i)-rhou(i-1))/(x(i)-x(i-1))
! sR=(rhou(i+1)-rhou(i))
sR=(rhou(i+1)-rhou(i))/(x(i+1)-x(i))
sL=sL
sR=sR
sC=LIMITER(sL,sR)
rhouR(i)=rhou(i)-sC*(0.5_KND)*(x(i)-x(i-1))
rhouL(i+1)=rhou(i)+sC*(0.5_KND)*(x(i+1)-x(i))
enddo
do i=0,nx+1
! sL=(e(i)-e(i-1))
sL=(e(i)-e(i-1))/(x(i)-x(i-1))
! sR=(e(i+1)-e(i))
sR=(e(i+1)-e(i))/(x(i+1)-x(i))
sL=sL
sR=sR
sC=LIMITER(sL,sR)
eR(i)=e(i)-sC*(0.5_KND)*(x(i)-x(i-1))
eL(i+1)=e(i)+sC*(0.5_KND)*(x(i+1)-x(i))
enddo
endsubroutine MUSCL
! Linear Piecewise extrapolation
subroutine PCONST(rhoL,rhouL,eL,rhoR,rhouR,eR,rho,rhou,e)
real(KND),dimension(-2:)::rhoL,rhouL,eL,rhoR,rhouR,eR,rho,rhou,e
integer i
do i=0,nx+1
rhoR(i)=rho(i)
rhoL(i+1)=rho(i)
enddo
do i=0,nx+1
rhouR(i)=rhou(i)
rhouL(i+1)=rhou(i)
enddo
do i=0,nx+1
eR(i)=e(i)
eL(i+1)=e(i)
enddo
endsubroutine PCONST
subroutine FLUXESLF(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR)
real(KND),dimension(-2:)::rhoL,rhouL,eL,rhoR,rhouR,eR
real(KND),dimension(LBOUND(rhoL,1):UBOUND(rhoL,1))::drho,drhou,de,pL,pR
real(KND) p,A
integer i
do i=1,nx+1
pL(i)=(gamma-1._KND)*(eL(i)-(1._KND/2._KND)*(rhouL(i)*rhouL(i))/rhoL(i))
pR(i)=(gamma-1._KND)*(eR(i)-(1._KND/2._KND)*(rhouR(i)*rhouR(i))/rhoR(i))
if ((pL(i)<0).or.(pR(i)<0)) write(*,*) "ERROR! pressure < 0",pL(i),pR(i)
enddo
A=dx/(dt)
!
!flux of rho
!
drho(1)=-(frho(rhoL(1),rhouL(1),eL(1),pL(1))+frho(rhoR(1),rhouR(1),eR(1),pR(1))&
-A*(rhoR(1)-rhoL(1)))/dx
do i=2,nx
p=frho(rhoL(i),rhouL(i),eL(i),pL(i))+frho(rhoR(i),rhouR(i),eR(i),pR(i))&
-A*(rhoR(i)-rhoL(i))
drho(i-1)=drho(i-1)+p/dx
drho(i)=drho(i)-p/dx
enddo
drho(nx)=drho(nx)+(frho(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(1))&
+frho(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(1))&
-A*(rhoR(nx+1)-rhoL(nx+1)))/dx
drho=-drho/2._KND
!
!flux of rho*u
!
drhou(1)=-(frhou(rhoL(1),rhouL(1),eL(1),pL(1))+frhou(rhoR(1),rhouR(1),eR(1),pR(1))&
-A*(rhouR(1)-rhouL(1)))/dx
do i=2,nx
p=frhou(rhoL(i),rhouL(i),eL(i),pL(i))+frhou(rhoR(i),rhouR(i),eR(i),pR(i))&
-A*(rhouR(i)-rhouL(i))
drhou(i-1)=drhou(i-1)+p/dx
drhou(i)=drhou(i)-p/dx
enddo
drhou(nx)=drhou(nx)+(frhou(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
+frhou(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1))&
-A*(rhouR(nx+1)-rhouL(nx+1)))/dx
drhou=-drhou/2._KND
!
!flux ef energy
!
de(1)=-(fe(rhoL(1),rhouL(1),eL(1),pL(1))+fe(rhoR(1),rhouR(1),eR(1),pR(1))&
-A*(eR(1)-eL(1)))/dx
do i=2,nx
p=fe(rhoL(i),rhouL(i),eL(i),pL(i))+fe(rhoR(i),rhouR(i),eR(i),pR(i))&
-A*(eR(i)-eL(i))
de(i-1)=de(i-1)+p/dx
de(i)=de(i)-p/dx
enddo
de(nx)=de(nx)+(fe(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
+fe(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1))&
-A*(eR(nx+1)-eL(nx+1)))/dx
de=-de/2._KND
endsubroutine FLUXESLF
subroutine FLUXESLW(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR)
real(KND),dimension(-2:)::rhoL,rhouL,eL,rhoR,rhouR,eR
real(KND),dimension(LBOUND(rhoL,1):UBOUND(rhoL,1))::drho,drhou,de,flrho,flrhou,fle
real(KND) A,pL,pR,pM
real(KND),DIMENSION(1:3):: qL,qR,qM,fl,fr
integer i
real(KND),PARAMETER:: eps=0.0001
A=dx/dt
do i=1,nx+1
qL=(/ rhoL(i),rhouL(i),eL(i) /)
qR=(/ rhoR(i),rhouR(i),eR(i) /)
pL=(gamma-1._KND)*(eL(i)-0.5_KND*(rhouL(i)*rhouL(i))/rhoL(i))
pR=(gamma-1._KND)*(eR(i)-0.5_KND*(rhouR(i)*rhouR(i))/rhoR(i))
if (pL<0) pL=0
if (pR<0) pR=0
fl(1)=frho(qL(1),qL(2),qL(3),pL)
fl(2)=frhou(qL(1),qL(2),qL(3),pL)
fl(3)=fe(qL(1),qL(2),qL(3),pL)
fr(1)=frho(qR(1),qR(2),qR(3),pR)
fr(2)=frhou(qR(1),qR(2),qR(3),pR)
fr(3)=fe(qR(1),qR(2),qR(3),pR)
qM=0.5_KND*(qL+qR)-(0.5_KND/A)*(fr-fl)
if (qM(1)<eps) qM(1)=eps
if (qM(3)<eps) qM(3)=eps
pM=(gamma-1._KND)*(qM(3)-0.5_KND*(qM(2)*qM(2))/qM(1))
if (pM<0) pM=0
flrho(i)=frho(qM(1),qM(2),qM(3),pM)
flrhou(i)=frhou(qM(1),qM(2),qM(3),pM)
fle(i)=fe(qM(1),qM(2),qM(3),pM)
enddo
drho(1)=flrho(1)/dx
drhou(1)=flrhou(1)/dx
de(1)=fle(1)/dx
do i=2,nx
drho(i-1)=drho(i-1)-flrho(i)/dx
drhou(i-1)=drhou(i-1)-flrhou(i)/dx
de(i-1)=de(i-1)-fle(i)/dx
drho(i)=flrho(i)/dx
drhou(i)=flrhou(i)/dx
de(i)=fle(i)/dx
enddo
drho(nx)=drho(nx)-flrho(nx+1)/dx
drhou(nx)=drhou(nx)-flrhou(nx+1)/dx
de(nx)=de(nx)-fle(nx+1)/dx
endsubroutine FLUXESLW
subroutine FLUXESKT(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR,x) !!!!!!!!!!!!!!!!!!!!KURGANOV TADMOR SCHEME !!!!!!!!!!!!!!!!!!!!!!!!!!!!!
real(KND),dimension(-2:)::rhoL,rhouL,eL,rhoR,rhouR,eR,x
real(KND),dimension(LBOUND(rhoL,1):UBOUND(rhoL,1))::drho,drhou,de,A,pL,pR
real(KND) p,cL,cR,dxnotu
integer i
do i=1,nx+1
pL(i)=(gamma-1._KND)*(eL(i)-(1._KND/2._KND)*(rhouL(i)*rhouL(i))/rhoL(i))
pR(i)=(gamma-1._KND)*(eR(i)-(1._KND/2._KND)*(rhouR(i)*rhouR(i))/rhoR(i))
if ((pL(i)<0).or.(pR(i)<0)) write(*,*) "ERROR! pressure < 0",pL(i),pR(i), &
eL(i),(1._KND/2._KND)*(rhouL(i)*rhouL(i))/rhoL(i),eR(i),(1._KND/2._KND)*(rhouR(i)*rhouR(i))/rhoR(i)
cL=SQRT(gamma*pL(i)/rhoL(i))
cR=SQRT(gamma*pR(i)/rhoR(i))
A(i)=MAX(ABS(rhouL(1)/rhoL(1)-cL),ABS(rhouR(1)/rhoR(1)-cR),&
ABS(rhouL(1)/rhoL(1)+cL),ABS(rhouR(1)/rhoR(1)+cR))
enddo
drho = 0.d0 !PER CHIAREZZA
drhou = 0.d0
de = 0.d0
!
!flux of rho
!
! drho(1)=-(frho(rhoL(1),rhouL(1),eL(1),pL(1))+frho(rhoR(1),rhouR(1),eR(1),pR(1))&
! -A(1)*(rhoR(1)-rhoL(1)))/dx
drho(1)=-(frho(rhoL(1),rhouL(1),eL(1),pL(1))+frho(rhoR(1),rhouR(1),eR(1),pR(1))&
-A(1)*(rhoR(1)-rhoL(1)))
do i=2,nx
p=frho(rhoL(i),rhouL(i),eL(i),pL(i))+frho(rhoR(i),rhouR(i),eR(i),pR(i))&
-A(i)*(rhoR(i)-rhoL(i))
! drho(i-1)=drho(i-1)+p/dx
drho(i-1)=drho(i-1)+p
! drho(i)=drho(i)-p/dx
drho(i)=drho(i)-p
enddo
! drho(nx)=drho(nx)+(frho(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(1))&
! +frho(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(1))&
! -A(nx+1)*(rhoR(nx+1)-rhoL(nx+1)))/dx
! drho=-drho/2._KND
drho(nx)=drho(nx)+(frho(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(1))&
+frho(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(1))&
-A(nx+1)*(rhoR(nx+1)-rhoL(nx+1)))
do i=1,nx
dxnotu = (x(i+1)-x(i-1))
drho(i) = -drho(i)/dxnotu
enddo
!
!flux of rho*u
!
! drhou(1)=-(frhou(rhoL(1),rhouL(1),eL(1),pL(1))+frhou(rhoR(1),rhouR(1),eR(1),pR(1))&
! -A(1)*(rhouR(1)-rhouL(1)))/dx
drhou(1)=-(frhou(rhoL(1),rhouL(1),eL(1),pL(1))+frhou(rhoR(1),rhouR(1),eR(1),pR(1))&
-A(1)*(rhouR(1)-rhouL(1)))
do i=2,nx
p=frhou(rhoL(i),rhouL(i),eL(i),pL(i))+frhou(rhoR(i),rhouR(i),eR(i),pR(i))&
-A(i)*(rhouR(i)-rhouL(i))
! drhou(i-1)=drhou(i-1)+p/dx
drhou(i-1)=drhou(i-1)+p
! drhou(i)=drhou(i)-p/dx
drhou(i)=drhou(i)-p
enddo
! drhou(nx)=drhou(nx)+(frhou(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
! +frhou(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1))&
! -A(nx+1)*(rhouR(nx+1)-rhouL(nx+1)))/dx
! drhou=-drhou/2._KND
drhou(nx)=drhou(nx)+(frhou(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
+frhou(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1))&
-A(nx+1)*(rhouR(nx+1)-rhouL(nx+1)))
do i=1,nx
dxnotu = (x(i+1)-x(i-1))
drhou(i) = -drhou(i)/dxnotu
enddo
!
!flux ef energy
!
! de(1)=-(fe(rhoL(1),rhouL(1),eL(1),pL(1))+fe(rhoR(1),rhouR(1),eR(1),pR(1))&
! -A(1)*(eR(1)-eL(1)))/dx
de(1)=-(fe(rhoL(1),rhouL(1),eL(1),pL(1))+fe(rhoR(1),rhouR(1),eR(1),pR(1))&
-A(1)*(eR(1)-eL(1)))
do i=2,nx
p=fe(rhoL(i),rhouL(i),eL(i),pL(i))+fe(rhoR(i),rhouR(i),eR(i),pR(i))&
-A(i)*(eR(i)-eL(i))
! de(i-1)=de(i-1)+p/dx
! de(i)=de(i)-p/dx
de(i-1)=de(i-1)+p
de(i)=de(i)-p
enddo
! de(nx)=de(nx)+(fe(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
! +fe(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1))&
! -A(nx+1)*(eR(nx+1)-eL(nx+1)))/dx
! de=-de/2._KND
de(nx)=de(nx)+(fe(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
+fe(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1))&
-A(nx+1)*(eR(nx+1)-eL(nx+1)))
do i=1,nx
dxnotu = (x(i+1)-x(i-1))
de(i) = -de(i)/dxnotu
enddo
endsubroutine FLUXESKT
subroutine FLUXESCU(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR)
real(KND),dimension(-2:)::rhoL,rhouL,eL,rhoR,rhouR,eR
real(KND),dimension(LBOUND(rhoL,1):UBOUND(rhoL,1))::drho,drhou,de,aL,aR,pL,pR
real(KND) cL,cR,p
integer i
do i=1,nx+1
pL(i)=(gamma-1._KND)*(eL(i)-(1._KND/2._KND)*(rhouL(i)*rhouL(i))/rhoL(i))
pR(i)=(gamma-1._KND)*(eR(i)-(1._KND/2._KND)*(rhouR(i)*rhouR(i))/rhoR(i))
if ((pL(i)<0).or.(pR(i)<0)) then
write(*,*) "ERROR! pressure < 0",pL(i),pR(i)
endif
cL=SQRT(gamma*pL(i)/rhoL(i))
cR=SQRT(gamma*pR(i)/rhoR(i))
aL(i)=MIN(rhouL(i)/rhoL(i)-cL,rhouR(i)/rhoR(i)-cR,0._KND)
aR(i)=MAX(rhouL(i)/rhoL(i)+cL,rhouR(i)/rhoR(i)+cR,0._KND)
enddo
!
!flux of rho
!
drho(1)=-((aR(1)*frho(rhoL(1),rhouL(1),eL(1),pL(1))&
-aL(1)*frho(rhoR(1),rhouR(1),eR(1),pR(1)))/(aR(1)-aL(1))&
+(aL(1)*aR(1)/(aR(1)-aL(1)))*(rhoR(1)-rhoL(1)))/dx
do i=2,nx
p=(aR(i)*frho(rhoL(i),rhouL(i),eL(i),pL(i))&
-aL(i)*frho(rhoR(i),rhouR(i),eR(i),pR(i)))/(aR(i)-aL(i))&
+(aL(i)*aR(i)/(aR(i)-aL(i)))*(rhoR(i)-rhoL(i))
drho(i-1)=drho(i-1)+p/dx
drho(i)=drho(i)-p/dx
enddo
drho(nx)=drho(nx)+((aR(nx+1)*frho(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
-aL(nx+1)*frho(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1)))/(aR(nx+1)-aL(nx+1))&
+(aL(nx+1)*aR(nx+1)/(aR(nx+1)-aL(nx+1)))*(rhoR(nx+1)-rhoL(nx+1)))/dx
drho=-drho
!
!flux of rho*u
!
drhou(1)=-((aR(1)*frhou(rhoL(1),rhouL(1),eL(1),pL(1))&
-aL(1)*frhou(rhoR(1),rhouR(1),eR(1),pR(1)))/(aR(1)-aL(1))&
+(aL(1)*aR(1)/(aR(1)-aL(1)))*(rhouR(1)-rhouL(1)))/dx
do i=2,nx
p=(aR(i)*frhou(rhoL(i),rhouL(i),eL(i),pL(i))&
-aL(i)*frhou(rhoR(i),rhouR(i),eR(i),pR(i)))/(aR(i)-aL(i))&
+(aL(i)*aR(i)/(aR(i)-aL(i)))*(rhouR(i)-rhouL(i))
drhou(i-1)=drhou(i-1)+p/dx
drhou(i)=drhou(i)-p/dx
enddo
drhou(nx)=drhou(nx)+((aR(nx+1)*frhou(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
-aL(nx+1)*frhou(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1)))/(aR(nx+1)-aL(nx+1))&
+(aL(nx+1)*aR(nx+1)/(aR(nx+1)-aL(nx+1)))*(rhouR(nx+1)-rhouL(nx+1)))/dx
drhou=-drhou
!
!flux ef energy
!
de(1)=-((aR(1)*fe(rhoL(1),rhouL(1),eL(1),pL(1))&
-aL(1)*fe(rhoR(1),rhouR(1),eR(1),pR(1)))/(aR(1)-aL(1))&
+(aL(1)*aR(1)/(aR(1)-aL(1)))*(eR(1)-eL(1)))/dx
do i=2,nx
p=(aR(i)*fe(rhoL(i),rhouL(i),eL(i),pL(i))&
-aL(i)*fe(rhoR(i),rhouR(i),eR(i),pR(i)))/(aR(i)-aL(i))&
+(aL(i)*aR(i)/(aR(i)-aL(i)))*(eR(i)-eL(i))
de(i-1)=de(i-1)+p/dx
de(i)=de(i)-p/dx
enddo
de(nx)=de(nx)+((aR(nx+1)*fe(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
-aL(nx+1)*fe(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1)))/(aR(nx+1)-aL(nx+1))&
+(aL(nx+1)*aR(nx+1)/(aR(nx+1)-aL(nx+1)))*(eR(nx+1)-eL(nx+1)))/dx
de=-de
endsubroutine FLUXESCU
subroutine FLUXESCULD(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR) !by Donato Cecere
real(KND),dimension(-2:)::rhoL,rhouL,eL,rhoR,rhouR,eR
real(KND),dimension(LBOUND(rhoL,1):UBOUND(rhoL,1))::drho,drhou,de,aL,aR,pL,pR
real(KND) cL,cR,p,sL,sR,wint,adt
integer i
do i=1,nx+1
pL(i)=(gamma-1._KND)*(eL(i)-(1._KND/2._KND)*(rhouL(i)*rhouL(i))/rhoL(i))
pR(i)=(gamma-1._KND)*(eR(i)-(1._KND/2._KND)*(rhouR(i)*rhouR(i))/rhoR(i))
if ((pL(i)<0).or.(pR(i)<0)) then
write(*,*) "ERROR! pressure < 0",pL(i),pR(i)
endif
cL=SQRT(gamma*pL(i)/rhoL(i))
cR=SQRT(gamma*pR(i)/rhoR(i))
aL(i)=MIN(rhouL(i)/rhoL(i)-cL,rhouR(i)/rhoR(i)-cR,0._KND)
aR(i)=MAX(rhouL(i)/rhoL(i)+cL,rhouR(i)/rhoR(i)+cR,0._KND)
enddo
!
!flux of rho
!
! 1
wint = (aR(1)*rhoR(1) -&
aL(1)*rhoL(1)-(frho(rhoR(1),rhouR(1),eR(1),pR(1)) -&
frho(rhoL(1),rhouL(1),eL(1),pL(1))))/(aR(1)-aL(1))
sR = (rhoR(1) - wint)/(aR(1)-aL(1))
sL = (wint - rhoL(1))/(aR(1)-aL(1))
adt = LIMITER(sL,sR)
drho(1)=-((aR(1)*frho(rhoL(1),rhouL(1),eL(1),pL(1))&
-aL(1)*frho(rhoR(1),rhouR(1),eR(1),pR(1)))/(aR(1)-aL(1))&
+(aL(1)*aR(1))*((rhoR(1)-rhoL(1))/(aR(1)-aL(1)) - adt))/dx
do i=2,nx
wint = (aR(i)*rhoR(i) -&
aL(i)*rhoL(i)-(frho(rhoR(i),rhouR(i),eR(i),pR(i)) -&
frho(rhoL(i),rhouL(i),eL(i),pL(i))))/(aR(i)-aL(i))
sR = (rhoR(i) - wint)/(aR(i)-aL(i))
sL = (wint - rhoL(i))/(aR(i)-aL(i))
adt = LIMITER(sL,sR)
p =(aR(i)*frho(rhoL(i),rhouL(i),eL(i),pL(i))&
-aL(i)*frho(rhoR(i),rhouR(i),eR(i),pR(i)))/(aR(i)-aL(i))&
+(aL(i)*aR(i)*( (rhoR(i)-rhoL(i))/(aR(i)-aL(i)) - adt ))
drho(i-1)=drho(i-1)+p/dx
drho(i)=drho(i)-p/dx
enddo
! nx
wint = (aR(nx)*rhoR(nx) -&
aL(nx)*rhoL(nx)-(frho(rhoR(nx),rhouR(nx),eR(nx),pR(nx)) -&
frho(rhoL(nx),rhouL(nx),eL(nx),pL(nx))))/(aR(nx)-aL(nx))
sR = (rhoR(nx) - wint)/(aR(nx)-aL(nx))
sL = (wint - rhoL(nx))/(aR(nx)-aL(nx))
adt = LIMITER(sL,sR)
drho(nx)=drho(nx)+((aR(nx+1)*frho(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
-aL(nx+1)*frho(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1)))/(aR(nx+1)-aL(nx+1))&
+(aL(nx+1)*aR(nx+1))*((rhoR(nx+1)-rhoL(nx+1))/(aR(nx+1)-aL(nx+1)) - adt))/dx
drho=-drho
!
!flux of rho*u
!
! 1
wint = (aR(1)*rhouR(1) -&
aL(1)*rhouL(1)-(frhou(rhoR(1),rhouR(1),eR(1),pR(1)) -&
frhou(rhoL(1),rhouL(1),eL(1),pL(1))))/(aR(1)-aL(1))
sR = (rhouR(1) - wint)/(aR(1)-aL(1))
sL = (wint - rhouL(1))/(aR(1)-aL(1))
adt = LIMITER(sL,sR)
drhou(1)=-((aR(1)*frhou(rhoL(1),rhouL(1),eL(1),pL(1))&
-aL(1)*frhou(rhoR(1),rhouR(1),eR(1),pR(1)))/(aR(1)-aL(1))&
+(aL(1)*aR(1))*((rhouR(1)-rhouL(1))/(aR(1)-aL(1)) - adt))/dx
do i=2,nx
wint = (aR(i)*rhouR(i) -&
aL(i)*rhouL(i)-(frhou(rhoR(i),rhouR(i),eR(i),pR(i)) -&
frhou(rhoL(i),rhouL(i),eL(i),pL(i))))/(aR(i)-aL(i))
sR = (rhouR(i) - wint)/(aR(i)-aL(i))
sL = (wint - rhouL(i))/(aR(i)-aL(i))
adt = LIMITER(sL,sR)
p=(aR(i)*frhou(rhoL(i),rhouL(i),eL(i),pL(i))&
-aL(i)*frhou(rhoR(i),rhouR(i),eR(i),pR(i)))/(aR(i)-aL(i))&
+(aL(i)*aR(i))*((rhouR(i)-rhouL(i))/(aR(i)-aL(i)) - adt)
drhou(i-1)=drhou(i-1)+p/dx
drhou(i)=drhou(i)-p/dx
enddo
! nx
wint = (aR(nx)*rhouR(nx) -&
aL(nx)*rhouL(nx)-(frhou(rhoR(nx),rhouR(nx),eR(nx),pR(nx)) -&
frhou(rhoL(nx),rhouL(nx),eL(nx),pL(nx))))/(aR(nx)-aL(nx))
sR = (rhouR(nx) - wint)/(aR(nx)-aL(nx))
sL = (wint - rhouL(nx))/(aR(i)-aL(nx))
adt = LIMITER(sL,sR)
drhou(nx)=drhou(nx)+((aR(nx+1)*frhou(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
-aL(nx+1)*frhou(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1)))/(aR(nx+1)-aL(nx+1))&
+(aL(nx+1)*aR(nx+1))*((rhouR(nx+1)-rhouL(nx+1))/(aR(nx+1)-aL(nx+1))))/dx
drhou=-drhou
!
!flux ef energy
!
! 1
wint = (aR(1)*rhouR(1) - aL(1)*rhouL(1)-(fe(rhoR(1),rhouR(1),eR(1),pR(1)) - fe(rhoL(1),rhouL(1),eL(1),pL(1))))/(aR(1)-aL(1))
sR = (rhouR(1) - wint)/(aR(1)-aL(1))
sL = (wint - rhouL(1))/(aR(1)-aL(1))
adt = LIMITER(sL,sR)
de(1)=-((aR(1)*fe(rhoL(1),rhouL(1),eL(1),pL(1))&
-aL(1)*fe(rhoR(1),rhouR(1),eR(1),pR(1)))/(aR(1)-aL(1))&
+(aL(1)*aR(1))*((eR(1)-eL(1))/(aR(1)-aL(1)) - adt))/dx
do i=2,nx
wint = (aR(i)*rhouR(i) - aL(i)*rhouL(i)-(fe(rhoR(i),rhouR(i),eR(i),pR(i)) - fe(rhoL(i),rhouL(i),eL(i),pL(i))))/(aR(i)-aL(i))
sR = (rhouR(i) - wint)/(aR(i)-aL(i))
sL = (wint - rhouL(i))/(aR(i)-aL(i))
adt = LIMITER(sL,sR)
p=(aR(i)*fe(rhoL(i),rhouL(i),eL(i),pL(i))&
-aL(i)*fe(rhoR(i),rhouR(i),eR(i),pR(i)))/(aR(i)-aL(i))&
+(aL(i)*aR(i))*((eR(i)-eL(i))/(aR(i)-aL(i)) - adt)
de(i-1)=de(i-1)+p/dx
de(i)=de(i)-p/dx
enddo
! nx
wint = (aR(nx)*rhouR(nx) -&
aL(nx)*rhouL(nx)-(fe(rhoR(nx),rhouR(nx),eR(nx),pR(nx)) -&
fe(rhoL(nx),rhouL(nx),eL(nx),pL(nx))))/(aR(nx)-aL(nx))
sR = (rhouR(nx) - wint)/(aR(nx)-aL(nx))
sL = (wint - rhouL(nx))/(aR(nx)-aL(nx))
adt = LIMITER(sL,sR)
de(nx)=de(nx)+((aR(nx+1)*fe(rhoL(nx+1),rhouL(nx+1),eL(nx+1),pL(nx+1))&
-aL(nx+1)*fe(rhoR(nx+1),rhouR(nx+1),eR(nx+1),pR(nx+1)))/(aR(nx+1)-aL(nx+1))&
+(aL(nx+1)*aR(nx+1))*(eR(nx+1)-eL(nx+1))/(aR(nx+1)-aL(nx+1)) - adt)/dx
de=-de
endsubroutine FLUXESCULD
subroutine FLUXESMUSTA(drho,drhou,de,rhoL,rhouL,eL,rhoR,rhouR,eR) !MUSTA
real(KND),dimension(-2:)::rhoL,rhouL,eL,rhoR,rhouR,eR
real(KND),dimension(LBOUND(rhoL,1):UBOUND(rhoL,1))::drho,drhou,de,&
flrho,flrhou,fle
real(KND) p,A,pL,pR,pM
real(KND),dimension(1:3):: qL,qR,qM,fL,fR,fM
integer i,l
A=dx/dt
do i=1,nx+1
stg: do l=1,MUSTAstages
if (l==1) then
qL=(/ rhoL(i),rhouL(i),eL(i) /)
qR=(/ rhoR(i),rhouR(i),eR(i) /)
pL=(gamma-1._KND)*(eL(i)-0.5_KND*(rhouL(i)*rhouL(i))/rhoL(i))
pR=(gamma-1._KND)*(eR(i)-0.5_KND*(rhouR(i)*rhouR(i))/rhoR(i))
endif
!predictor step
fl(1)=frho(qL(1),qL(2),qL(3),pL)
fl(2)=frhou(qL(1),qL(2),qL(3),pL)
fl(3)=fe(qL(1),qL(2),qL(3),pL)
fr(1)=frho(qR(1),qR(2),qR(3),pR)
fr(2)=frhou(qR(1),qR(2),qR(3),pR)
fr(3)=fe(qR(1),qR(2),qR(3),pR)
qM=0.5_KND*(qL+qR)-(0.5_KND/A)*(fr-fl)
pM=(gamma-1._KND)*(qM(3)-0.5_KND*(qM(2)*qM(2))/qM(1))
fM(1)=frho(qM(1),qM(2),qM(3),pM)
fM(2)=frhou(qM(1),qM(2),qM(3),pM)
fM(3)=fe(qM(1),qM(2),qM(3),pM)
fM=0.25_KND*(fL+2*fM+fR-A*(qR-qL))
if (l==MUSTAstages) then
flrho(i)=fM(1)
flrhou(i)=fM(2)
fle(i)=fM(3)
EXIT stg
else
!corrector step - opens Riemann fan
qL=qL-(fM-fL)/A
qR=qR-(fR-fM)/A
pL=(gamma-1._KND)*(qL(3)-0.5_KND*(qL(2)*qL(2))/qL(1))
pR=(gamma-1._KND)*(qR(3)-0.5_KND*(qR(2)*qR(2))/qR(1))
endif
enddo stg
enddo
!update differences
drho(1)=flrho(1)/dx
drhou(1)=flrhou(1)/dx
de(1)=fle(1)/dx
do i=2,nx
drho(i-1)=drho(i-1)-flrho(i)/dx
drhou(i-1)=drhou(i-1)-flrhou(i)/dx
de(i-1)=de(i-1)-fle(i)/dx
drho(i)=flrho(i)/dx
drhou(i)=flrhou(i)/dx
de(i)=fle(i)/dx
enddo
drho(nx)=drho(nx)-flrho(nx+1)/dx
drhou(nx)=drhou(nx)-flrhou(nx+1)/dx
de(nx)=de(nx)-fle(nx+1)/dx
endsubroutine FLUXESMUSTA
subroutine FDLF(rho,rhou,e)
real(KND),dimension(-2:)::rho,rhou,e
real(KND),dimension(LBOUND(rho,1):UBOUND(rho,1))::p,rho2,rhou2,e2
real(KND) A
integer i
call BOUND(rho,rhou,e)
do i=0,nx+1
p(i)=(gamma-1._KND)*(e(i)-(1._KND/2._KND)*(rhou(i)*rhou(i))/rho(i))
if ((p(i)<0).or.(p(i)<0)) write(*,*) "ERROR! pressure < 0",p(i)
enddo
A=(dt/dx)
do i=1,nx
rho2(i)=(rho(i-1)+rho(i+1))/2._KND&
-A*(frho(rho(i+1),rhou(i+1),e(i+1),p(i+1))-frho(rho(i-1),rhou(i-1),e(i-1),p(i-1)))/2._KND
rhou2(i)=(rhou(i-1)+rhou(i+1))/2._KND&
-A*(frhou(rho(i+1),rhou(i+1),e(i+1),p(i+1))-frhou(rho(i-1),rhou(i-1),e(i-1),p(i-1)))/2._KND
e2(i)=(e(i-1)+e(i+1))/2._KND&
-A*(fe(rho(i+1),rhou(i+1),e(i+1),p(i+1))-fe(rho(i-1),rhou(i-1),e(i-1),p(i-1)))/2._KND
enddo
rho=rho2
rhou=rhou2
e=e2
endsubroutine FDLF
subroutine FDLW(rho,rhou,e)
real(KND),dimension(-2:)::rho,rhou,e
real(KND),dimension(LBOUND(rho,1):UBOUND(rho,1))::p,rho2,rhou2,e2
real(KND) A
integer i
real(KND),PARAMETER:: eps=0.001
call BOUND(rho,rhou,e)
do i=0,nx+1
p(i)=(gamma-1._KND)*(e(i)-(1._KND/2._KND)*(rhou(i)*rhou(i))/rho(i))
if (p(i)<0) p=0
enddo
A=dt/dx
do i=1,nx
rho2(i)=(rho(i)+rho(i+1))/2._KND&
-A*(frho(rho(i+1),rhou(i+1),e(i+1),p(i+1))-frho(rho(i),rhou(i),e(i),p(i)))/2._KND
if (rho2(i)<eps) rho2(i)=eps
rhou2(i)=(rhou(i)+rhou(i+1))/2._KND&
-A*(frhou(rho(i+1),rhou(i+1),e(i+1),p(i+1))-frhou(rho(i),rhou(i),e(i),p(i)))/2._KND
e2(i)=(e(i)+e(i+1))/2._KND&
-A*(fe(rho(i+1),rhou(i+1),e(i+1),p(i+1))-fe(rho(i),rhou(i),e(i),p(i)))/2._KND
if (e2(i)<eps) e2(i)=eps
enddo
call BOUND(rho2,rhou2,e2)
do i=0,nx+1
p(i)=(gamma-1._KND)*(e2(i)-(1._KND/2._KND)*(rhou2(i)*rhou2(i))/rho2(i))
if (p(i)<0) p=0
enddo
do i=1,nx
rho(i)=rho(i)-A*(frho(rho2(i),rhou2(i),e2(i),p(i))-frho(rho2(i-1),rhou2(i-1),e2(i-1),p(i-1)))
if (rho(i)<eps) rho(i)=eps
rhou(i)=rhou(i)-A*(frhou(rho2(i),rhou2(i),e2(i),p(i))-frhou(rho2(i-1),rhou2(i-1),e2(i-1),p(i-1)))
e(i)=e(i)-A*(fe(rho2(i),rhou2(i),e2(i),p(i))-fe(rho2(i-1),rhou2(i-1),e2(i-1),p(i-1)))
if (e(i)<eps) e(i)=eps
enddo
endsubroutine FDLW
real(KND) function frho(rho,rhou,e,p)
real(KND) rho,rhou,e,p
frho=rhou
endfunction frho
real(KND) function frhou(rho,rhou,e,p)
real(KND) rho,rhou,e,p
frhou=rhou*rhou/rho+p
endfunction frhou
real(KND) function fe(rho,rhou,e,p)
real(KND) rho,rhou,e,p
fe=(rhou/rho)*(e+p)
endfunction fe
real(KND) function LIMITER(a,b)
real(KND) a,b
if (limitertype==1) then
LIMITER=MINMOD(a,b)
elseif (limitertype==2) then
LIMITER=MINMOD(limparam*a,(a+b)/2._KND,limparam*b)
elseif (limitertype==3) then
LIMITER=SUPERBEE(a,b)
endif
endfunction LIMITER
real(KND) function MINMOD2(a,b)
real(KND) a,b
MINMOD2=(SIGN(1._KND,a)+SIGN(1._KND,b))*MIN(ABS(a),ABS(b))/2._KND
endfunction MINMOD2
real(KND) function MINMOD3(a,b,c)
real(KND) a,b,c
if ((a>0._KND).and.(b>0._KND).and.(c>0._KND)) then
MINMOD3=MIN(a,b,c)
elseif ((a<0._KND).and.(b<0._KND).and.(c<0._KND)) then
MINMOD3=MAX(a,b,c)
else
MINMOD3=0._KND
endif
endfunction MINMOD3
real(KND) function SUPERBEE(a,b)
real(KND) a,b,ab,b2
ab=a*b
b2=b*b
if ((a*b)<=0._KND) then
SUPERBEE=0._KND
else if (limparam*ab<b2) then
SUPERBEE=a*limparam
else if (ab<b2) then
SUPERBEE=b
else if (ab<limparam*b2) then
SUPERBEE=a
else
SUPERBEE=b*limparam
endif
endfunction SUPERBEE
subroutine BOUND(rho,rhou,e)
real(KND),dimension(-2:)::rho,rhou,e
if (extype==1) then
rho(-2:0)=rho(1)
rho(nx+1:nx+3)=rho(nx)
rhou(-2:0)=rhou(1)
rhou(nx+1:nx+3)=rhou(nx)
e(-2:0)=e(1)
e(nx+1:nx+3)=e(nx)
else
rho(-2:0)=rho(nx-2:nx)
rho(nx+1:nx+3)=rho(1:3)
rhou(-2:0)=rhou(nx-2:nx)
rhou(nx+1:nx+3)=rhou(1:3)
e(-2:0)=e(nx-2:nx)
e(nx+1:nx+3)=e(1:3)
endif
endsubroutine BOUND
integer function lowerIndex(xvalue, x, nx)
integer nx,i
real(KND) x(nx)
real(KND) xvalue
!print*,' '
do i=2,nx-1
if (xvalue < x(i)) exit
end do
lowerIndex = i-1
return
endfunction lowerIndex
subroutine linearMesh(yp, r, n, y, ny)
!
! yp locations of segment matching points (returns xp)
! r relative stretching to be applied at matching points
! n number of matching points (including end points)
! y mesh coordinates (output)
! ny number of mesh points
integer n,ny,i,j
real(KND) yp(n), r(n)
real(KND) y(ny)
real(KND)xp(n)
real(KND)a,b,x
xp(1) = 0.
do i=2,n
a = r(i-1)
b = (r(i) - r(i-1)) / (yp(i) - yp(i-1))
if (b == 0) then
xp(i) = xp(i-1) + (1./a)*(yp(i) - yp(i-1))
else
xp(i) = xp(i-1) + (1./b)*log((a + b*(yp(i) - yp(i-1))) / a)
end if
end do
do j=1,ny
x = (j-1.)/(ny-1.)*xp(n)
i = lowerIndex(x, xp, n)
a = r(i)
b = (r(i+1) - r(i)) / (yp(i+1) - yp(i))
if (b == 0) then
y(j) = yp(i) + a*(x - xp(i))
else
y(j) = yp(i) + (a/b)*(exp(b*(x - xp(i))) - 1.)
end if
end do
! Normalize xp-coordinates
xp = xp / xp(n)
! Return xp-coordinates in yp
yp = xp
return
endsubroutine linearMesh
endmodule CUEuler