c
c
c =====================================================
subroutine rpn2eu(ixy,maxm,meqn,mwaves,mbc,mx,ql,qr,maux,
& auxl,auxr,wave,s,amdq,apdq)
c =====================================================
c
c # Roe-solver for the Euler equations
c # solve Riemann problems along one slice of data.
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
c # This data is along a slice in the x-direction if ixy=1
c # or the y-direction if ixy=2.
c # On output, wave contains the waves, s the speeds,
c # and amdq, apdq the decomposition of the flux difference
c # f(qr(i-1)) - f(ql(i))
c # into leftgoing and rightgoing parts respectively.
c # With the Roe solver we have
c # amdq = A^- \Delta q and apdq = A^+ \Delta q
c # where A is the Roe matrix. An entropy fix can also be incorporated
c # into the flux differences.
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 routines, this routine is called with ql = qr
c
c Author: Randall J. LeVeque
c
implicit double precision (a-h,o-z)
dimension wave(1-mbc:maxm+mbc, meqn, mwaves)
dimension s(1-mbc:maxm+mbc, mwaves)
dimension ql(1-mbc:maxm+mbc, meqn)
dimension qr(1-mbc:maxm+mbc, meqn)
dimension apdq(1-mbc:maxm+mbc, meqn)
dimension amdq(1-mbc:maxm+mbc, meqn)
c
c local arrays -- common block comroe is passed to rpt2eu
c ------------
parameter (maxm2 = 10005) !# assumes at most 10000x10000 grid with mbc=5
parameter (minm2 = -4) !# assumes at most mbc=5
dimension delta(4)
logical efix
common /param/ gamma,gamma1
common /comroe/ u2v2(minm2:maxm2),
& u(minm2:maxm2),v(minm2:maxm2),enth(minm2:maxm2),
& a(minm2:maxm2),g1a2(minm2:maxm2),euv(minm2:maxm2)
c
data efix /.true./ !# use entropy fix for transonic rarefactions
c
c # Riemann solver returns flux differences
c ------------
common /rpnflx/ mrpnflx
mrpnflx = 0
c
if (minm2.gt.1-mbc .or. maxm2.lt.maxm+mbc) then
write(6,*) 'need to increase maxm2 in rpA'
stop
endif
c
c # set mu to point to the component of the system that corresponds
c # to momentum in the direction of this slice, mv to the orthogonal
c # momentum:
c
if (ixy.eq.1) then
mu = 2
mv = 3
else
mu = 3
mv = 2
endif
c
c # note that notation for u and v reflects assumption that the
c # Riemann problems are in the x-direction with u in the normal
c # direciton and v in the orthogonal direcion, but with the above
c # definitions of mu and mv the routine also works with ixy=2
c # and returns, for example, f0 as the Godunov flux g0 for the
c # Riemann problems u_t + g(u)_y = 0 in the y-direction.
c
c
c # compute the Roe-averaged variables needed in the Roe solver.
c # These are stored in the common block comroe since they are
c # later used in routine rpt2eu to do the transverse wave splitting.
c
do 10 i = 2-mbc, mx+mbc
rhsqrtl = dsqrt(qr(i-1,1))
rhsqrtr = dsqrt(ql(i,1))
pl = gamma1*(qr(i-1,4) - 0.5d0*(qr(i-1,2)**2 +
& qr(i-1,3)**2)/qr(i-1,1))
pr = gamma1*(ql(i,4) - 0.5d0*(ql(i,2)**2 +
& ql(i,3)**2)/ql(i,1))
rhsq2 = rhsqrtl + rhsqrtr
u(i) = (qr(i-1,mu)/rhsqrtl + ql(i,mu)/rhsqrtr) / rhsq2
v(i) = (qr(i-1,mv)/rhsqrtl + ql(i,mv)/rhsqrtr) / rhsq2
enth(i) = (((qr(i-1,4)+pl)/rhsqrtl
& + (ql(i,4)+pr)/rhsqrtr)) / rhsq2
u2v2(i) = u(i)**2 + v(i)**2
a2 = gamma1*(enth(i) - .5d0*u2v2(i))
a(i) = dsqrt(a2)
g1a2(i) = gamma1 / a2
euv(i) = enth(i) - u2v2(i)
10 continue
c
c
c # now split the jump in q at each interface into waves
c
c # find a1 thru a4, the coefficients of the 4 eigenvectors:
do 20 i = 2-mbc, mx+mbc
delta(1) = ql(i,1) - qr(i-1,1)
delta(2) = ql(i,mu) - qr(i-1,mu)
delta(3) = ql(i,mv) - qr(i-1,mv)
delta(4) = ql(i,4) - qr(i-1,4)
a3 = g1a2(i) * (euv(i)*delta(1)
& + u(i)*delta(2) + v(i)*delta(3) - delta(4))
a2 = delta(3) - v(i)*delta(1)
a4 = (delta(2) + (a(i)-u(i))*delta(1) - a(i)*a3) / (2.d0*a(i))
a1 = delta(1) - a3 - a4
c
c # Compute the waves.
c # Note that the 2-wave and 3-wave travel at the same speed and
c # are lumped together in wave(.,.,2). The 4-wave is then stored in
c # wave(.,.,3).
c
wave(i,1,1) = a1
wave(i,mu,1) = a1*(u(i)-a(i))
wave(i,mv,1) = a1*v(i)
wave(i,4,1) = a1*(enth(i) - u(i)*a(i))
s(i,1) = u(i)-a(i)
c
wave(i,1,2) = a3
wave(i,mu,2) = a3*u(i)
wave(i,mv,2) = a3*v(i) + a2
wave(i,4,2) = a3*0.5d0*u2v2(i) + a2*v(i)
s(i,2) = u(i)
c
wave(i,1,3) = a4
wave(i,mu,3) = a4*(u(i)+a(i))
wave(i,mv,3) = a4*v(i)
wave(i,4,3) = a4*(enth(i)+u(i)*a(i))
s(i,3) = u(i)+a(i)
20 continue
c
c
c # compute flux differences amdq and apdq.
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=1,4
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-1,4) - 0.5d0*(qr(i-1,mu)**2
& + qr(i-1,mv)**2) / rhoim1)
cim1 = dsqrt(gamma*pim1/rhoim1)
s0 = qr(i-1,mu)/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=1,4
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,mu) + wave(i,mu,1)
rhov1 = qr(i-1,mv) + wave(i,mv,1)
en1 = qr(i-1,4) + wave(i,4,1)
p1 = gamma1*(en1 - 0.5d0*(rhou1**2 + rhov1**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=1,4
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- and 3- waves are rightgoing
do 140 m=1,4
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,4) - 0.5d0*(ql(i,mu)**2
& + ql(i,mv)**2) / rhoi)
ci = dsqrt(gamma*pi/rhoi)
s3 = ql(i,mu)/rhoi + ci !# u+c in right state (cell i)
c
rho2 = ql(i,1) - wave(i,1,3)
rhou2 = ql(i,mu) - wave(i,mu,3)
rhov2 = ql(i,mv) - wave(i,mv,3)
en2 = ql(i,4) - wave(i,4,3)
p2 = gamma1*(en2 - 0.5d0*(rhou2**2 + rhov2**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=1,4
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=1,4
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
900 continue
return
end