c
c
c
c =========================================================
subroutine rp1eu(maxmx,meqn,mwaves,mbc,mx,ql,qr,maux,
& auxl,auxr,wave,s,amdq,apdq)
c =========================================================
c
c # solve Riemann problems for the 1D Euler equations using Roe's
c # approximate Riemann solver.
c
c # On input, ql contains the state vector at the left edge of each cell
c # qr contains the state vector at the right edge of each cell
c # On output, wave contains the waves,
c # s the speeds,
c # amdq the left-going flux difference A^- \Delta q
c # apdq the right-going flux difference A^+ \Delta q
c
c # Note that the i'th Riemann problem has left state qr(i-1,:)
c # and right state ql(i,:)
c # From the basic routine step1, rp is called with ql = qr = q.
c
c Author: Randall J. LeVeque
c
implicit double precision (a-h,o-z)
dimension ql(1-mbc:maxmx+mbc, meqn)
dimension qr(1-mbc:maxmx+mbc, meqn)
dimension s(1-mbc:maxmx+mbc, mwaves)
dimension wave(1-mbc:maxmx+mbc, meqn, mwaves)
dimension amdq(1-mbc:maxmx+mbc, meqn)
dimension apdq(1-mbc:maxmx+mbc, meqn)
c
c # local storage
c ---------------
parameter (max2 = 100002) !# assumes at most 100000 grid points with mbc=2
dimension delta(3)
dimension u(-1:max2),enth(-1:max2),a(-1:max2)
logical efix
common /param/ gamma,gamma1
c
data efix /.true./ !# use entropy fix for transonic rarefactions
c
c # Riemann solver returns flux differences
c ------------
common /rpnflx/ mrpnflx
mrpnflx = 0
c
c # Compute Roe-averaged quantities:
c
do 20 i=2-mbc,mx+mbc
rhsqrtl = dsqrt(qr(i-1,1))
rhsqrtr = dsqrt(ql(i,1))
pl = gamma1*(qr(i-1,3) - 0.5d0*(qr(i-1,2)**2)/qr(i-1,1))
pr = gamma1*(ql(i,3) - 0.5d0*(ql(i,2)**2)/ql(i,1))
rhsq2 = rhsqrtl + rhsqrtr
u(i) = (qr(i-1,2)/rhsqrtl + ql(i,2)/rhsqrtr) / rhsq2
enth(i) = (((qr(i-1,3)+pl)/rhsqrtl
& + (ql(i,3)+pr)/rhsqrtr)) / rhsq2
a2 = gamma1*(enth(i) - .5d0*u(i)**2)
a(i) = dsqrt(a2)
20 continue
c
c
do 30 i=2-mbc,mx+mbc
c
c # find a1 thru a3, the coefficients of the 3 eigenvectors:
c
delta(1) = ql(i,1) - qr(i-1,1)
delta(2) = ql(i,2) - qr(i-1,2)
delta(3) = ql(i,3) - qr(i-1,3)
a2 = gamma1/a(i)**2 * ((enth(i)-u(i)**2)*delta(1)
& + u(i)*delta(2) - delta(3))
a3 = (delta(2) + (a(i)-u(i))*delta(1) - a(i)*a2) / (2.d0*a(i))
a1 = delta(1) - a2 - a3
c
c # Compute the waves.
c
wave(i,1,1) = a1
wave(i,2,1) = a1*(u(i)-a(i))
wave(i,3,1) = a1*(enth(i) - u(i)*a(i))
s(i,1) = u(i)-a(i)
c
wave(i,1,2) = a2
wave(i,2,2) = a2*u(i)
wave(i,3,2) = a2*0.5d0*u(i)**2
s(i,2) = u(i)
c
wave(i,1,3) = a3
wave(i,2,3) = a3*(u(i)+a(i))
wave(i,3,3) = a3*(enth(i)+u(i)*a(i))
s(i,3) = u(i)+a(i)
30 continue
c
c # compute Godunov flux f0:
c --------------------------
c
c
if (efix) go to 110
c
c # no entropy fix
c ----------------
c
c # amdq = SUM s*wave over left-going waves
c # apdq = SUM s*wave over right-going waves
c
do 100 m=1,3
do 100 i=2-mbc, mx+mbc
amdq(i,m) = 0.d0
apdq(i,m) = 0.d0
do 90 mw=1,mwaves
if (s(i,mw) .lt. 0.d0) then
amdq(i,m) = amdq(i,m) + s(i,mw)*wave(i,m,mw)
else
apdq(i,m) = apdq(i,m) + s(i,mw)*wave(i,m,mw)
endif
90 continue
100 continue
go to 900
c
c-----------------------------------------------------
c
110 continue
c
c # With entropy fix
c ------------------
c
c # compute flux differences amdq and apdq.
c # First compute amdq as sum of s*wave for left going waves.
c # Incorporate entropy fix by adding a modified fraction of wave
c # if s should change sign.
c
do 200 i=2-mbc,mx+mbc
c
c # check 1-wave:
c ---------------
c
rhoim1 = qr(i-1,1)
pim1 = gamma1*(qr(i-1,3) - 0.5d0*qr(i-1,2)**2 / rhoim1)
cim1 = dsqrt(gamma*pim1/rhoim1)
s0 = qr(i-1,2)/rhoim1 - cim1 !# u-c in left state (cell i-1)
c # check for fully supersonic case:
if (s0.ge.0.d0 .and. s(i,1).gt.0.d0) then
c # everything is right-going
do 60 m=1,3
amdq(i,m) = 0.d0
60 continue
go to 200
endif
c
rho1 = qr(i-1,1) + wave(i,1,1)
rhou1 = qr(i-1,2) + wave(i,2,1)
en1 = qr(i-1,3) + wave(i,3,1)
p1 = gamma1*(en1 - 0.5d0*rhou1**2/rho1)
c1 = dsqrt(gamma*p1/rho1)
s1 = rhou1/rho1 - c1 !# u-c to right of 1-wave
if (s0.lt.0.d0 .and. s1.gt.0.d0) then
c # transonic rarefaction in the 1-wave
sfract = s0 * (s1-s(i,1)) / (s1-s0)
else if (s(i,1) .lt. 0.d0) then
c # 1-wave is leftgoing
sfract = s(i,1)
else
c # 1-wave is rightgoing
sfract = 0.d0 !# this shouldn't happen since s0 < 0
endif
do 120 m=1,3
amdq(i,m) = sfract*wave(i,m,1)
120 continue
c
c # check 2-wave:
c ---------------
c
if (s(i,2) .ge. 0.d0) go to 200 !# 2-wave is rightgoing
do 140 m=1,3
amdq(i,m) = amdq(i,m) + s(i,2)*wave(i,m,2)
140 continue
c
c # check 3-wave:
c ---------------
c
rhoi = ql(i,1)
pi = gamma1*(ql(i,3) - 0.5d0*ql(i,2)**2 / rhoi)
ci = dsqrt(gamma*pi/rhoi)
s3 = ql(i,2)/rhoi + ci !# u+c in right state (cell i)
c
rho2 = ql(i,1) - wave(i,1,3)
rhou2 = ql(i,2) - wave(i,2,3)
en2 = ql(i,3) - wave(i,3,3)
p2 = gamma1*(en2 - 0.5d0*rhou2**2/rho2)
c2 = dsqrt(gamma*p2/rho2)
s2 = rhou2/rho2 + c2 !# u+c to left of 3-wave
if (s2 .lt. 0.d0 .and. s3.gt.0.d0) then
c # transonic rarefaction in the 3-wave
sfract = s2 * (s3-s(i,3)) / (s3-s2)
else if (s(i,3) .lt. 0.d0) then
c # 3-wave is leftgoing
sfract = s(i,3)
else
c # 3-wave is rightgoing
go to 200
endif
c
do 160 m=1,3
amdq(i,m) = amdq(i,m) + sfract*wave(i,m,3)
160 continue
200 continue
c
c # compute the rightgoing flux differences:
c # df = SUM s*wave is the total flux difference and apdq = df - amdq
c
do 220 m=1,3
do 220 i = 2-mbc, mx+mbc
df = 0.d0
do 210 mw=1,mwaves
df = df + s(i,mw)*wave(i,m,mw)
210 continue
apdq(i,m) = df - amdq(i,m)
220 continue
c
c
900 continue
c
return
end