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SCHEMES: Fix adiabatic HLLD solver for weak Bx.

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If the parallel component of magnetic field is too small, the
intermediate states might produce numerical instabilities, since they
are obtained by the division by a small factor.

In order to remove this situation, we apply full HLLD solver for strong
enough Alfvén wave. If it is weak, the HLLC solver is applied.

Signed-off-by: Grzegorz Kowal <grzegorz@amuncode.org>
This commit is contained in:
Grzegorz Kowal 2017-04-10 08:03:50 -03:00
parent 8dc157620c
commit da2f2a1bf0
1 changed files with 88 additions and 39 deletions
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127
src/schemes.F90
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@ -3668,35 +3668,35 @@ module schemes
dn = wr(idn) - wl(idn) dn = wr(idn) - wl(idn)
sm = (wr(imx) - wl(imx)) / dn sm = (wr(imx) - wl(imx)) / dn
! speed differences
!
slmm = sl - sm
srmm = sr - sm
! left and right state densities
!
dnl = wl(idn) / slmm
dnr = wr(idn) / srmm
! square of Bₓ, i.e. Bₓ² ! square of Bₓ, i.e. Bₓ²
! !
bx = ql(ibx,i) bx = ql(ibx,i)
b2 = ql(ibx,i) * qr(ibx,i) b2 = ql(ibx,i) * qr(ibx,i)
! if there is an Alvén wave, apply the full HLLD solver, otherwise revert to
! the HLLC solver
!
if (b2 > 0.0d+00) then ! Bₓ² > 0
! the total pressure, constant across the contact discontinuity and Alfvén waves ! the total pressure, constant across the contact discontinuity and Alfvén waves
! !
pt = (wl(idn) * wr(imx) - wr(idn) * wl(imx)) / dn + b2 pt = (wl(idn) * wr(imx) - wr(idn) * wl(imx)) / dn + b2
! speed differences ! if Alfvén wave is strong enough, apply the full HLLD solver, otherwise
! revert to the HLLC one
! !
slmm = sl - sm if (b2 > 1.0d-08 * max(dnl * sl**2, dnr * sr**2)) then ! Bₓ² > ε
srmm = sr - sm
! left and right state densities
!
dnl = wl(idn) / slmm
dnr = wr(idn) / srmm
! left and right Alfvén speeds ! left and right Alfvén speeds
! !
cal = - sqrt(b2 / dnl) cal = sqrt(b2 / dnl)
car = sqrt(b2 / dnr) car = sqrt(b2 / dnr)
sml = sm + cal sml = sm - cal
smr = sm + car smr = sm + car
! calculate division factors (also used to determine degeneracies) ! calculate division factors (also used to determine degeneracies)
@ -3814,10 +3814,10 @@ module schemes
! prepare constant primitive variables of the intermediate states ! prepare constant primitive variables of the intermediate states
! !
dv = car * dnr - cal * dnl dv = car * dnr + cal * dnl
vy = (wcr(imy) - wcl(imy)) / dv vy = (wcr(imy) - wcl(imy)) / dv
vz = (wcr(imz) - wcl(imz)) / dv vz = (wcr(imz) - wcl(imz)) / dv
dv = car - cal dv = car + cal
by = (wcr(iby) - wcl(iby)) / dv by = (wcr(iby) - wcl(iby)) / dv
bz = (wcr(ibz) - wcl(ibz)) / dv bz = (wcr(ibz) - wcl(ibz)) / dv
vb = sm * bx + vy * by + vz * bz vb = sm * bx + vy * by + vz * bz
@ -3837,7 +3837,7 @@ module schemes
ui(iby) = by ui(iby) = by
ui(ibz) = bz ui(ibz) = bz
ui(ibp) = ul(ibp,i) ui(ibp) = ul(ibp,i)
ui(ien) = (wcl(ien) + sm * pt - bx * vb) / cal ui(ien) = (wcl(ien) + sm * pt - bx * vb) / (sml - sm)
! the inmost left intermediate flux ! the inmost left intermediate flux
! !
@ -3855,7 +3855,7 @@ module schemes
ui(iby) = by ui(iby) = by
ui(ibz) = bz ui(ibz) = bz
ui(ibp) = ur(ibp,i) ui(ibp) = ur(ibp,i)
ui(ien) = (wcr(ien) + sm * pt - bx * vb) / car ui(ien) = (wcr(ien) + sm * pt - bx * vb) / (smr - sm)
! the inmost right intermediate flux ! the inmost right intermediate flux
! !
@ -3914,7 +3914,7 @@ module schemes
! primitive variables for the inner left intermediate state ! primitive variables for the inner left intermediate state
! !
dv = srmm * dnr - cal * dnl dv = srmm * dnr + cal * dnl
vy = (wr(imy) - wcl(imy)) / dv vy = (wr(imy) - wcl(imy)) / dv
vz = (wr(imz) - wcl(imz)) / dv vz = (wr(imz) - wcl(imz)) / dv
dv = sr - sml dv = sr - sml
@ -3932,7 +3932,7 @@ module schemes
ui(iby) = by ui(iby) = by
ui(ibz) = bz ui(ibz) = bz
ui(ibp) = ul(ibp,i) ui(ibp) = ul(ibp,i)
ui(ien) = (wcl(ien) + sm * pt - bx * vb) / cal ui(ien) = (wcl(ien) + sm * pt - bx * vb) / (sml - sm)
! choose the correct state depending on the sign of contact discontinuity ! choose the correct state depending on the sign of contact discontinuity
! advection speed ! advection speed
@ -4013,7 +4013,7 @@ module schemes
ui(iby) = by ui(iby) = by
ui(ibz) = bz ui(ibz) = bz
ui(ibp) = ur(ibp,i) ui(ibp) = ur(ibp,i)
ui(ien) = (wcr(ien) + sm * pt - bx * vb) / car ui(ien) = (wcr(ien) + sm * pt - bx * vb) / (smr - sm)
! choose the correct state depending on the sign of contact discontinuity ! choose the correct state depending on the sign of contact discontinuity
! advection speed ! advection speed
@ -4048,24 +4048,77 @@ module schemes
end if ! one degeneracy end if ! one degeneracy
else ! Bₓ² = 0 else if (b2 > 0.0d+00) then ! Bₓ² > 0
! the total pressure, constant across the contact discontinuity ! constant intermediate state tangential components of velocity and
! magnetic field
! !
pt = (wl(idn) * wr(imx) - wr(idn) * wl(imx)) / dn vy = (wr(imy) - wl(imy)) / dn
vz = (wr(imz) - wl(imz)) / dn
by = (wr(iby) - wl(iby)) / srml
bz = (wr(ibz) - wl(ibz)) / srml
! scalar product of velocity and magnetic field for the intermediate states
!
vb = sm * bx + vy * by + vz * bz
! separate intermediate states depending on the sign of the advection speed ! separate intermediate states depending on the sign of the advection speed
! !
if (sm > 0.0d+00) then ! sm > 0 if (sm > 0.0d+00) then ! sm > 0
! the left speed difference ! conservative variables for the left intermediate state
! !
slmm = sl - sm ui(idn) = dnl
ui(imx) = dnl * sm
ui(imy) = dnl * vy
ui(imz) = dnl * vz
ui(ibx) = bx
ui(iby) = by
ui(ibz) = bz
ui(ibp) = ul(ibp,i)
ui(ien) = (wl(ien) + sm * pt - bx * vb) / slmm
! the left intermediate flux
!
f(:,i) = sl * ui(:) - wl(:)
else if (sm < 0.0d+00) then ! sm < 0
! conservative variables for the right intermediate state
!
ui(idn) = dnr
ui(imx) = dnr * sm
ui(imy) = dnr * vy
ui(imz) = dnr * vz
ui(ibx) = bx
ui(iby) = by
ui(ibz) = bz
ui(ibp) = ur(ibp,i)
ui(ien) = (wr(ien) + sm * pt - bx * vb) / srmm
! the right intermediate flux
!
f(:,i) = sr * ui(:) - wr(:)
else ! sm = 0
! when Sₘ = 0 all variables are continuous, therefore the flux reduces
! to the HLL one
!
f(:,i) = (sl * wr(:) - sr * wl(:)) / srml
end if ! sm = 0
else ! Bₓ² = 0
! separate intermediate states depending on the sign of the advection speed
!
if (sm > 0.0d+00) then ! sm > 0
! conservative variables for the left intermediate state ! conservative variables for the left intermediate state
! !
ui(idn) = wl(idn) / slmm ui(idn) = dnl
ui(imx) = ui(idn) * sm ui(imx) = dnl * sm
ui(imy) = wl(imy) / slmm ui(imy) = wl(imy) / slmm
ui(imz) = wl(imz) / slmm ui(imz) = wl(imz) / slmm
ui(ibx) = 0.0d+00 ui(ibx) = 0.0d+00
@ -4080,14 +4133,10 @@ module schemes
else if (sm < 0.0d+00) then ! sm < 0 else if (sm < 0.0d+00) then ! sm < 0
! the right speed difference
!
srmm = sr - sm
! conservative variables for the right intermediate state ! conservative variables for the right intermediate state
! !
ui(idn) = wr(idn) / srmm ui(idn) = dnr
ui(imx) = ui(idn) * sm ui(imx) = dnr * sm
ui(imy) = wr(imy) / srmm ui(imy) = wr(imy) / srmm
ui(imz) = wr(imz) / srmm ui(imz) = wr(imz) / srmm
ui(ibx) = 0.0d+00 ui(ibx) = 0.0d+00
@ -4102,8 +4151,8 @@ module schemes
else ! sm = 0 else ! sm = 0
! intermediate flux is constant across the contact discontinuity, so we end up ! when Sₘ = 0 all variables are continuous, therefore the flux reduces
! having only one intermediate state, which is equivalent to the HLL solver ! to the HLL one
! !
f(:,i) = (sl * wr(:) - sr * wl(:)) / srml f(:,i) = (sl * wr(:) - sr * wl(:)) / srml