在 Cygwin 终端中，我输入
$ gfortran -o threed_euler_fluxes_v3.exe threed_euler_fluxes_v3.f90
我得到编译器错误
/usr/lib/gcc/i686-pc-cygwin/4.5.3/../../../libcygwin.a(libcmain.o): In function `main':
/usr/src/debug/cygwin-1.7.17-1/winsup/cygwin/lib/libcmain.c:39: undefined reference to `_WinMain@16'
collect2: ld returned 1 exit status
我也试过这样编译
$ gfortran -o threed_euler_fluxes_v3.exe threed_euler_fluxes_v3.f90 -shared
但是当我尝试运行时，我得到一个错误，说它不是一个有效的 Windows 程序？
这是完整的fortran代码。我删除了一些评论，以便将字数限制保持在 30k 以下。这是原文。
subroutine inviscid_roe(primL, primR, njk, num_flux)
implicit none
integer , parameter :: p2 = selected_real_kind(15) ! Double precision
!Input
real(p2), intent( in) :: primL(5), primR(5) ! Input: primitive variables
real(p2), intent( in) :: njk(3) ! Input: face normal vector
!Output
real(p2), intent(out) :: num_flux(5) ! Output: numerical flux
!Some constants
real(p2) :: zero = 0.0_p2
real(p2) :: one = 1.0_p2
real(p2) :: two = 2.0_p2
real(p2) :: half = 0.5_p2
real(p2) :: fifth = 0.2_p2
!Local variables
real(p2) :: nx, ny, nz ! Normal vector
real(p2) :: mx, my, mz ! Orthogonal tangent vector
real(p2) :: lx, ly, lz ! Another orthogonal tangent vector
real(p2) :: abs_n_cross_l ! Magnitude of n x l
real(p2) :: uL, uR, vL, vR, wL, wR ! Velocity components.
real(p2) :: rhoL, rhoR, pL, pR ! Primitive variables.
real(p2) :: qnL, qnR, qmL, qmR, qlL, qlR ! Normal and tangent velocities
real(p2) :: aL, aR, HL, HR ! Speed of sound, Total enthalpy
real(p2) :: RT,rho,u,v,w,H,a,qn, ql, qm ! Roe-averages
real(p2) :: drho,dqn,dql,dqm,dp,LdU(5) ! Wave strengths
real(p2) :: ws(5), R(5,5) ! Wave speeds and right-eigenvectors
real(p2) :: dws(5) ! Width of a parabolic fit for entropy fix
real(p2) :: fL(5), fR(5), diss(5) ! Fluxes ad dissipation term
real(p2) :: gamma = 1.4_p2 ! Ratio of specific heats
real(p2) :: temp, tempx, tempy, tempz ! Temoprary variables
! Face normal vector (unit vector)
nx = njk(1)
ny = njk(2)
nz = njk(3)
tempx = ny*ny + nz*nz
tempy = nz*nz + nx*nx
tempz = nx*nx + ny*ny
if ( tempx >= tempy .and. tempx >= tempz ) then
lx = zero
ly = -nz
lz = ny
elseif ( tempy >= tempx .and. tempy >= tempz ) then
lx = -nz
ly = zero
lz = nx
elseif ( tempz >= tempx .and. tempz >= tempy ) then
lx = -ny
ly = nx
lz = zero
else
! Impossible to happen
write(*,*) "subroutine inviscid_roe: Impossible to happen. Please report the problem."
stop
endif
! Make it the unit vector.
temp = sqrt( lx*lx + ly*ly + lz*lz )
lx = lx/temp
ly = ly/temp
lz = lz/temp
mx = ny*lz - nz*ly
my = nz*lx - nx*lz
mz = nx*ly - ny*lx
abs_n_cross_l = sqrt(mx**2 + my**2 + mz**2)
mx = mx / abs_n_cross_l
my = my / abs_n_cross_l
mz = mz / abs_n_cross_l
rhoL = primL(1)
uL = primL(2)
vL = primL(3)
wL = primL(4)
qnL = uL*nx + vL*ny + wL*nz
qlL = uL*lx + vL*ly + wL*lz
qmL = uL*mx + vL*my + wL*mz
pL = primL(5)
aL = sqrt(gamma*pL/rhoL)
HL = aL*aL/(gamma-one) + half*(uL*uL+vL*vL+wL*wL)
! Right state
rhoR = primR(1)
uR = primR(2)
vR = primR(3)
wR = primR(4)
qnR = uR*nx + vR*ny + wR*nz
qlR = uR*lx + vR*ly + wR*lz
qmR = uR*mx + vR*my + wR*mz
pR = primR(5)
aR = sqrt(gamma*pR/rhoR)
HR = aR*aR/(gamma-one) + half*(uR*uR+vR*vR+wR*wR)
RT = sqrt(rhoR/rhoL)
rho = RT*rhoL !Roe-averaged density
u = (uL + RT*uR)/(one + RT) !Roe-averaged x-velocity
v = (vL + RT*vR)/(one + RT) !Roe-averaged y-velocity
w = (wL + RT*wR)/(one + RT) !Roe-averaged z-velocity
H = (HL + RT*HR)/(one + RT) !Roe-averaged total enthalpy
a = sqrt( (gamma-one)*(H-half*(u*u + v*v + w*w)) ) !Roe-averaged speed of sound
qn = u*nx + v*ny + w*nz !Roe-averaged face-normal velocity
ql = u*lx + v*ly + w*lz !Roe-averaged face-tangent velocity
qm = u*mx + v*my + w*mz !Roe-averaged face-tangent velocity
!Wave Strengths
drho = rhoR - rhoL !Density difference
dp = pR - pL !Pressure difference
dqn = qnR - qnL !Normal velocity difference
dql = qlR - qlL !Tangent velocity difference in l
dqm = qmR - qmL !Tangent velocity difference in m
LdU(1) = (dp - rho*a*dqn )/(two*a*a) !Left-moving acoustic wave strength
LdU(2) = drho - dp/(a*a) !Entropy wave strength
LdU(3) = (dp + rho*a*dqn )/(two*a*a) !Right-moving acoustic wave strength
LdU(4) = rho*dql !Shear wave strength
LdU(5) = rho*dqm !Shear wave strength
!Absolute values of the wave speeds
ws(1) = abs(qn-a) !Left-moving acoustic wave speed
ws(2) = abs(qn) !Entropy wave speed
ws(3) = abs(qn+a) !Right-moving acoustic wave speed
ws(4) = abs(qn) !Shear wave speed
ws(5) = abs(qn) !Shear wave speed
!Harten's Entropy Fix JCP(1983), 49, pp357-393: only for the nonlinear fields.
!NOTE: It avoids vanishing wave speeds by making a parabolic fit near ws = 0.
dws(1) = fifth
if ( ws(1) < dws(1) ) ws(1) = half * ( ws(1)*ws(1)/dws(1)+dws(1) )
dws(3) = fifth
if ( ws(3) < dws(3) ) ws(3) = half * ( ws(3)*ws(3)/dws(3)+dws(3) )
!Right Eigenvectors
! Left-moving acoustic wave
R(1,1) = one
R(2,1) = u - a*nx
R(3,1) = v - a*ny
R(4,1) = w - a*nz
R(5,1) = H - a*qn
! Entropy wave
R(1,2) = one
R(2,2) = u
R(3,2) = v
R(4,2) = w
R(5,2) = half*(u*u + v*v + w*w)
! Right-moving acoustic wave
R(1,3) = one
R(2,3) = u + a*nx
R(3,3) = v + a*ny
R(4,3) = w + a*nz
R(5,3) = H + a*qn
! Shear wave
R(1,4) = zero
R(2,4) = lx
R(3,4) = ly
R(4,4) = lz
R(5,4) = ql
! Shear wave
R(1,5) = zero
R(2,5) = mx
R(3,5) = my
R(4,5) = mz
R(5,5) = qm
diss(:) = ws(1)*LdU(1)*R(:,1) + ws(2)*LdU(2)*R(:,2) + ws(3)*LdU(3)*R(:,3) &
+ ws(4)*LdU(4)*R(:,4) + ws(5)*LdU(5)*R(:,5)
fL(1) = rhoL*qnL
fL(2) = rhoL*qnL * uL + pL*nx
fL(3) = rhoL*qnL * vL + pL*ny
fL(4) = rhoL*qnL * wL + pL*nz
fL(5) = rhoL*qnL * HL
fR(1) = rhoR*qnR
fR(2) = rhoR*qnR * uR + pR*nx
fR(3) = rhoR*qnR * vR + pR*ny
fR(4) = rhoR*qnR * wR + pR*nz
fR(5) = rhoR*qnR * HR
num_flux = half * (fL + fR - diss)
subroutine inviscid_roe_n(primL, primR, njk, num_flux)
implicit none
integer , parameter :: p2 = selected_real_kind(15) ! Double precision
!Input
real(p2), intent( in) :: primL(5), primR(5) ! Input: primitive variables
real(p2), intent( in) :: njk(3) ! Input: face normal vector
!Output
real(p2), intent(out) :: num_flux(5) ! Output: numerical flux
!Some constants
real(p2) :: zero = 0.0_p2
real(p2) :: one = 1.0_p2
real(p2) :: two = 2.0_p2
real(p2) :: half = 0.5_p2
real(p2) :: fifth = 0.2_p2
!Local variables
real(p2) :: nx, ny, nz ! Normal vector
real(p2) :: uL, uR, vL, vR, wL, wR ! Velocity components.
real(p2) :: rhoL, rhoR, pL, pR ! Primitive variables.
real(p2) :: qnL, qnR ! Normal velocities
real(p2) :: aL, aR, HL, HR ! Speed of sound, Total enthalpy
real(p2) :: RT,rho,u,v,w,H,a,qn ! Roe-averages
real(p2) :: drho,dqn,dp,LdU(4) ! Wave strengths
real(p2) :: du, dv, dw ! Velocity differences
real(p2) :: ws(4), R(5,4) ! Wave speeds and right-eigenvectors
real(p2) :: dws(4) ! Width of a parabolic fit for entropy fix
real(p2) :: fL(5), fR(5), diss(5) ! Fluxes ad dissipation term
real(p2) :: gamma = 1.4_p2 ! Ratio of specific heats
! Face normal vector (unit vector)
nx = njk(1)
ny = njk(2)
nz = njk(3)
!Primitive and other variables.
! Left state
rhoL = primL(1)
uL = primL(2)
vL = primL(3)
wL = primL(4)
qnL = uL*nx + vL*ny + wL*nz
pL = primL(5)
aL = sqrt(gamma*pL/rhoL)
HL = aL*aL/(gamma-one) + half*(uL*uL+vL*vL+wL*wL)
! Right state
rhoR = primR(1)
uR = primR(2)
vR = primR(3)
wR = primR(4)
qnR = uR*nx + vR*ny + wR*nz
pR = primR(5)
aR = sqrt(gamma*pR/rhoR)
HR = aR*aR/(gamma-one) + half*(uR*uR+vR*vR+wR*wR)
!First compute the Roe-averaged quantities
! NOTE: See http://www.cfdnotes.com/cfdnotes_roe_averaged_density.html for
! the Roe-averaged density.
RT = sqrt(rhoR/rhoL)
rho = RT*rhoL !Roe-averaged density
u = (uL + RT*uR)/(one + RT) !Roe-averaged x-velocity
v = (vL + RT*vR)/(one + RT) !Roe-averaged y-velocity
w = (wL + RT*wR)/(one + RT) !Roe-averaged z-velocity
H = (HL + RT*HR)/(one + RT) !Roe-averaged total enthalpy
a = sqrt( (gamma-one)*(H-half*(u*u + v*v + w*w)) ) !Roe-averaged speed of sound
qn = u*nx + v*ny + w*nz !Roe-averaged face-normal velocity
!Wave Strengths
drho = rhoR - rhoL !Density difference
dp = pR - pL !Pressure difference
dqn = qnR - qnL !Normal velocity difference
LdU(1) = (dp - rho*a*dqn )/(two*a*a) !Left-moving acoustic wave strength
LdU(2) = drho - dp/(a*a) !Entropy wave strength
LdU(3) = (dp + rho*a*dqn )/(two*a*a) !Right-moving acoustic wave strength
LdU(4) = rho !Shear wave strength (not really, just a factor)
!Absolute values of the wave Speeds
ws(1) = abs(qn-a) !Left-moving acoustic wave
ws(2) = abs(qn) !Entropy wave
ws(3) = abs(qn+a) !Right-moving acoustic wave
ws(4) = abs(qn) !Shear waves
!Harten's Entropy Fix JCP(1983), 49, pp357-393: only for the nonlinear fields.
!NOTE: It avoids vanishing wave speeds by making a parabolic fit near ws = 0.
dws(1) = fifth
if ( ws(1) < dws(1) ) ws(1) = half * ( ws(1)*ws(1)/dws(1)+dws(1) )
dws(3) = fifth
if ( ws(3) < dws(3) ) ws(3) = half * ( ws(3)*ws(3)/dws(3)+dws(3) )
R(1,1) = one
R(2,1) = u - a*nx
R(3,1) = v - a*ny
R(4,1) = w - a*nz
R(5,1) = H - a*qn
R(1,2) = one
R(2,2) = u
R(3,2) = v
R(4,2) = w
R(5,2) = half*(u*u + v*v + w*w)
! Right-moving acoustic wave
R(1,3) = one
R(2,3) = u + a*nx
R(3,3) = v + a*ny
R(4,3) = w + a*nz
R(5,3) = H + a*qn
! Two shear wave components combined into one (wave strength incorporated).
du = uR - uL
dv = vR - vL
dw = wR - wL
R(1,4) = zero
R(2,4) = du - dqn*nx
R(3,4) = dv - dqn*ny
R(4,4) = dw - dqn*nz
R(5,4) = u*du + v*dv + w*dw - qn*dqn
!Dissipation Term: |An|(UR-UL) = R|Lambda|L*dU = sum_k of [ ws(k) * R(:,k) * L*dU(k) ]
diss(:) = ws(1)*LdU(1)*R(:,1) + ws(2)*LdU(2)*R(:,2) &
+ ws(3)*LdU(3)*R(:,3) + ws(4)*LdU(4)*R(:,4)
!Compute the physical flux: fL = Fn(UL) and fR = Fn(UR)
fL(1) = rhoL*qnL
fL(2) = rhoL*qnL * uL + pL*nx
fL(3) = rhoL*qnL * vL + pL*ny
fL(4) = rhoL*qnL * wL + pL*nz
fL(5) = rhoL*qnL * HL
fR(1) = rhoR*qnR
fR(2) = rhoR*qnR * uR + pR*nx
fR(3) = rhoR*qnR * vR + pR*ny
fR(4) = rhoR*qnR * wR + pR*nz
fR(5) = rhoR*qnR * HR
! This is the numerical flux: Roe flux = 1/2 *[ Fn(UL)+Fn(UR) - |An|(UR-UL) ]
num_flux = half * (fL + fR - diss)
!Normal max wave speed in the normal direction.
! wsn = abs(qn) + a
end subroutine inviscid_roe_n
subroutine inviscid_rotated_rhll(primL, primR, njk, num_flux)
implicit none
integer , parameter :: p2 = selected_real_kind(15) ! Double precision
!Input
real(p2), intent( in) :: primL(5), primR(5) ! Input: primitive variables
real(p2), intent( in) :: njk(3) ! Input: face normal vector
!Output
real(p2), intent(out) :: num_flux(5) ! Output: numerical flux
!Some constants
real(p2) :: zero = 0.0_p2
real(p2) :: one = 1.0_p2
real(p2) :: two = 2.0_p2
real(p2) :: half = 0.5_p2
real(p2) :: fifth = 0.2_p2
!Local variables
real(p2) :: nx, ny, nz ! Face normal vector
real(p2) :: uL, uR, vL, vR, wL, wR ! Velocity components.
real(p2) :: rhoL, rhoR, pL, pR ! Primitive variables.
real(p2) :: qnL, qnR ! Normal velocity
real(p2) :: aL, aR, HL, HR ! Speed of sound, Total enthalpy
real(p2) :: RT,rho,u,v,w,H,a,qn ! Roe-averages
real(p2) :: drho,dqn,dp,LdU(4) ! Wave strengths
real(p2) :: du, dv, dw ! Velocity conponent differences
real(p2) :: eig(4) ! Eigenvalues
real(p2) :: ws(4), R(5,4) ! Absolute Wave speeds and right-eigenvectors
real(p2) :: dws(4) ! Width of a parabolic fit for entropy fix
real(p2) :: fL(5), fR(5), diss(5) ! Fluxes ad dissipation term
real(p2) :: gamma = 1.4_p2 ! Ratio of specific heats
real(p2) :: SRp,SLm ! Wave speeds for the HLL part
real(p2) :: nx1, ny1, nz1 ! Vector along which HLL is applied
real(p2) :: nx2, ny2, nz2 ! Vector along which Roe is applied
real(p2) :: alpha1, alpha2 ! Projections of the new normals
real(p2) :: abs_dq ! Magnitude of the velocity difference
real(p2) :: temp, tempx, tempy, tempz ! Temporary variables
! Face normal vector (unit vector)
nx = njk(1)
ny = njk(2)
nz = njk(3)
!Primitive and other variables.
! Left state
rhoL = primL(1)
uL = primL(2)
vL = primL(3)
wL = primL(4)
qnL = uL*nx + vL*ny + wL*nz
pL = primL(5)
aL = sqrt(gamma*pL/rhoL)
HL = aL*aL/(gamma-one) + half*(uL*uL+vL*vL+wL*wL)
! Right state
rhoR = primR(1)
uR = primR(2)
vR = primR(3)
wR = primR(4)
qnR = uR*nx + vR*ny + wR*nz
pR = primR(5)
aR = sqrt(gamma*pR/rhoR)
HR = aR*aR/(gamma-one) + half*(uR*uR+vR*vR+wR*wR)
!Compute the physical flux: fL = Fn(UL) and fR = Fn(UR)
fL(1) = rhoL*qnL
fL(2) = rhoL*qnL * uL + pL*nx
fL(3) = rhoL*qnL * vL + pL*ny
fL(4) = rhoL*qnL * wL + pL*nz
fL(5) = rhoL*qnL * HL
fR(1) = rhoR*qnR
fR(2) = rhoR*qnR * uR + pR*nx
fR(3) = rhoR*qnR * vR + pR*ny
fR(4) = rhoR*qnR * wR + pR*nz
fR(5) = rhoR*qnR * HR
abs_dq = sqrt( (uR-uL)**2 + (vR-vL)**2 + (wR-wL)**2 )
if ( abs_dq > 1.0e-12_p2) then
nx1 = (uR-uL)/abs_dq
ny1 = (vR-vL)/abs_dq
nz1 = (wR-wL)/abs_dq
tempx = ny*ny + nz*nz
tempy = nz*nz + nx*nx
tempz = nx*nx + ny*ny
if ( tempx >= tempy .and. tempx >= tempz ) then
nx1 = zero
ny1 = -nz
nz1 = ny
elseif ( tempy >= tempx .and. tempy >= tempz ) then
nx1 = -nz
ny1 = zero
nz1 = nx
elseif ( tempz >= tempx .and. tempz >= tempy ) then
nx1 = -ny
ny1 = nx
nz1 = zero
else
! Impossible to happen
write(*,*) "inviscid_rotated_rhll: Impossible to happen. Please report the problem."
stop
endif
! Make it the unit vector.
temp = sqrt( nx1*nx1 + ny1*ny1 + nz1*nz1 )
nx1 = nx1/temp
ny1 = ny1/temp
nz1 = nz1/temp
endif
alpha1 = nx*nx1 + ny*ny1 + nz*nz1
! Make alpha1 always positive.
temp = sign(one,alpha1)
nx1 = temp * nx1
ny1 = temp * ny1
nz1 = temp * nz1
alpha1 = temp * alpha1
!n2 = direction perpendicular to n1.
! Note: There are infinitely many choices for this vector.
! The best choice may be discovered in future.
! Here, we employ the formula (4.4) in the paper:
! (nx2,ny2,nz2) = (n1xn)xn1 / |(n1xn)xn1| ('x' is the vector product.)
! (tempx,tempy,tempz) = n1xn
tempx = ny1*nz - nz1*ny
tempy = nz1*nx - nx1*nz
tempz = nx1*ny - ny1*nx
! (nx2,ny2,nz2) = (n1xn)xn1
nx2 = tempy*nz1 - tempz*ny1
ny2 = tempz*nx1 - tempx*nz1
nz2 = tempx*ny1 - tempy*nx1
! Make n2 the unit vector
temp = sqrt( nx2*nx2 + ny2*ny2 + nz2*nz2 )
nx2 = nx2/temp
ny2 = ny2/temp
nz2 = nz2/temp
alpha2 = nx*nx2 + ny*ny2 + nz*nz2
! Make alpha2 always positive.
temp = sign(one,alpha2)
nx2 = temp * nx2
ny2 = temp * ny2
nz2 = temp * nz2
alpha2 = temp * alpha2
!--------------------------------------------------------------------------------
!Now we are going to compute the Roe flux with n2 as the normal with modified
!wave speeds (5.12). NOTE: the Roe flux here is computed without tangent vectors.
!See "I do like CFD, VOL.1" for details: page 57, Equation (3.6.31).
!First compute the Roe-averaged quantities
! NOTE: See http://www.cfdnotes.com/cfdnotes_roe_averaged_density.html for
! the Roe-averaged density.
RT = sqrt(rhoR/rhoL)
rho = RT*rhoL !Roe-averaged density.
u = (uL + RT*uR)/(one + RT) !Roe-averaged x-velocity
v = (vL + RT*vR)/(one + RT) !Roe-averaged y-velocity
w = (wL + RT*wR)/(one + RT) !Roe-averaged z-velocity
H = (HL + RT*HR)/(one + RT) !Roe-averaged total enthalpy
a = sqrt( (gamma-one)*(H-half*(u*u + v*v + w*w)) ) !Roe-averaged speed of sound
!----------------------------------------------------
!Compute the wave speed estimates for the HLL part,
!following Einfeldt:
!
! B. Einfeldt, On Godunov-type methods for gas dynamics,
! SIAM Journal on Numerical Analysis 25 (2) (1988) 294–318.
!
! Note: HLL is actually applied to n1, but this is
! all we need to incorporate HLL. See JCP2008 paper.
qn = u *nx1 + v *ny1 + w *nz1
qnL = uL*nx1 + vL*ny1 + wL*nz1
qnR = uR*nx1 + vR*ny1 + wR*nz1
SLm = min( zero, qn - a, qnL - aL ) !Minimum wave speed estimate
SRp = max( zero, qn + a, qnR + aR ) !Maximum wave speed estimate
! This is the only place where n1=(nx1,ny1,nz1) is used.
! n1=(nx1,ny1,nz1) is never used below.
!----------------------------------------------------
!Wave Strengths
qn = u *nx2 + v *ny2 + w *nz2
qnL = uL*nx2 + vL*ny2 + wL*nz2
qnR = uR*nx2 + vR*ny2 + wR*nz2
drho = rhoR - rhoL !Density difference
dp = pR - pL !Pressure difference
dqn = qnR - qnL !Normal velocity difference
LdU(1) = (dp - rho*a*dqn )/(two*a*a) !Left-moving acoustic wave strength
LdU(2) = drho - dp/(a*a) !Entropy wave strength
LdU(3) = (dp + rho*a*dqn )/(two*a*a) !Right-moving acoustic wave strength
LdU(4) = rho !Shear wave strength (not really, just a factor)
!Wave Speed (Eigenvalues)
eig(1) = qn-a !Left-moving acoustic wave velocity
eig(2) = qn !Entropy wave velocity
eig(3) = qn+a !Right-moving acoustic wave velocity
eig(4) = qn !Shear wave velocity
!Absolute values of the wave speeds (Eigenvalues)
ws(1) = abs(qn-a) !Left-moving acoustic wave speed
ws(2) = abs(qn) !Entropy wave speed
ws(3) = abs(qn+a) !Right-moving acoustic wave speed
ws(4) = abs(qn) !Shear wave speed
!Harten's Entropy Fix JCP(1983), 49, pp357-393: only for the nonlinear fields.
!NOTE: It avoids vanishing wave speeds by making a parabolic fit near ws = 0.
dws(1) = fifth
if ( ws(1) < dws(1) ) ws(1) = half * ( ws(1)*ws(1)/dws(1)+dws(1) )
dws(3) = fifth
if ( ws(3) < dws(3) ) ws(3) = half * ( ws(3)*ws(3)/dws(3)+dws(3) )
!Combine the wave speeds for Rotated-RHLL: Eq.(5.12) in the original JCP2008 paper.
ws = alpha2*ws - (alpha1*two*SRp*SLm + alpha2*(SRp+SLm)*eig)/(SRp-SLm)
!Below, we compute the Roe dissipation term in the direction n2
!with the above modified wave speeds. HLL wave speeds act something like
!the entropy fix or eigenvalue limiting; they contribute only by the amount
!given by the fraction, alpha1 (less than or equal to 1.0). See JCP2008 paper.
!Right Eigenvectors:
!Note: Two shear wave components are combined into one, so that tangent vectors
! are not required. And that's why there are only 4 vectors here.
! Left-moving acoustic wave
R(1,1) = one
R(2,1) = u - a*nx2
R(3,1) = v - a*ny2
R(4,1) = w - a*nz2
R(5,1) = H - a*qn
! Entropy wave
R(1,2) = one
R(2,2) = u
R(3,2) = v
R(4,2) = w
R(5,2) = half*(u*u + v*v + w*w)
! Right-moving acoustic wave
R(1,3) = one
R(2,3) = u + a*nx2
R(3,3) = v + a*ny2
R(4,3) = w + a*nz2
R(5,3) = H + a*qn
! Two shear wave components combined into one (wave strength incorporated).
du = uR - uL
dv = vR - vL
dw = wR - wL
R(1,4) = zero
R(2,4) = du - dqn*nx2
R(3,4) = dv - dqn*ny2
R(4,4) = dw - dqn*nz2
R(5,4) = u*du + v*dv + w*dw - qn*dqn
!Dissipation Term: Roe dissipation with the modified wave speeds.
! |An|dU = R|Lambda|L*dU = sum_k of [ ws(k) * R(:,k) * L*dU(k) ], where n=n2.
diss(:) = ws(1)*LdU(1)*R(:,1) + ws(2)*LdU(2)*R(:,2) &
+ ws(3)*LdU(3)*R(:,3) + ws(4)*LdU(4)*R(:,4)
!Compute the Rotated-RHLL flux. (It looks like the HLL flux with Roe dissipation.)
num_flux = (SRp*fL - SLm*fR)/(SRp-SLm) - half*diss
!Normal max wave speed in the normal direction.
! wsn = abs(qn) + a
end subroutine inviscid_rotated_rhll
!--------------------------------------------------------------------------------