c c c ===================================================== subroutine rpn2(ixy,maxm,meqn,mwaves,mbc,mx,ql,qr,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 clawpack routines, this routine is called with ql = qr c c implicit double precision (a-h,o-z) c 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 = 602) !# assumes at most 200x200 grid with mbc=2 dimension delta(4) logical efix common /param/ gamma,gamma1 common /comroe/ u2v2(-1:maxm2), & u(-1:maxm2),v(-1:maxm2),enth(-1:maxm2),a(-1:maxm2), & g1a2(-1:maxm2),euv(-1:maxm2) c data efix /.true./ !# use entropy fix for transonic rarefactions c if (-1.gt.1-mbc .or. maxm2 .lt. maxm+mbc) then write(6,*) 'need to increase maxm2 in rpn2' 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