With smoothed P, resolution 256^3, L=1\times10^{13} cm, and run time t=2\times10^6 seconds\approx 4 dynamical times, No velocity damping, unless otherwise indicated

I) Boundary conditions on the Poisson solver

Motivation:

• It was realized last meeting that "Multipole expansion" BCs on the Poisson solver might be more reasonable than periodic BCs.

• Therefore I did a run that was the same as the fiducial run from last blog post but with Multipole expansion.

Setup:
Simply changed the Poisson BCs in the global.data file.

Results:

a) Periodic BCs, no velocity damping
(i) Constant ambient pressure and density (Ambp038, 5.2 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION: Star expands and contracts, becoming first more diamond-shaped and then more square-shaped.

(ii) Isothermal hydrostatic atmosphere (Atm001, 4.1 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity
DESCRIPTION: Star becomes unstable after about 1 dynamical time.

b) Multipole expansion BCs, no velocity damping
(i) Constant ambient pressure and density (Damp010, 5 hrs on comet compute, 120 cores)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION: Star becomes square-shaped, starts to become unstable after a few dynamical times.

(ii) Isothermal hydrostatic atmosphere (Atm003)
​2d density ​2d density and velocity
DESCRIPTION: Star becomes unstable after about 1 dynamical time.

Comparison with (a) on left and (b) on right
(i) Constant ambient pressure and density
​2d density ​2d density and velocity

• Case (b) becomes boxy and goes unstable a bit earlier than case (a)

• No oscillations are present in (b), contrary to (a).

• Outside of the star, velocity magnitudes are similar in (a) and (b).

Conclusion
Multipole expansion (ME) BCs is more physical. It averts unphysical oscillations. Therefore, we adopt ME BCs below.

II) Boundary conditions on the hydrodynamical quantities

Motivation:

• With extrapolated BCs we obtain reflections at the boundary and inflows.

• We want to try other BCs to avert these reflections/inflows as much as possible.

Setup:

• In ProblemBeforeStep we try two alternative BCs, in turn:
  

1) Fix the boundaries to the values of \rho, P and v(=0) of the initial profile
or
2) Draw a sphere with radius L/2 and fix the points outside this sphere to the values of \rho, P, and v(=0) of the initial profile.

Results:

a) Extrapolated hydro BCs, Multipole expansion Poisson BCs, no velocity damping (same as (b) in Sect. I above)
(i) Constant ambient pressure and density (Damp010)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION: Star becomes square-shaped, and unstable after a few dynamical times.

(ii) Isothermal hydrostatic atmosphere (Atm003)
​2d density ​2d density and velocity
DESCRIPTION: Star becomes unstable after about 1 dynamical time.

b) Fixed profile (\rho, P, \bm{v}=0) on boundary hydro BCs, Multipole expansion Poisson BCs, no velocity damping
(i) Constant ambient pressure and density (Damp017, 9 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION: Star becomes square-shaped, starts to become unstable after a few dynamical times.

(ii) Isothermal hydrostatic atmosphere (Atm004)
​2d density ​2d density and velocity
DESCRIPTION: Star becomes unstable after about 1 dynamical time.

c) Fixed profile outside sphere (of radius 5\times10^{12} cm) hydro BCs, Multipole expansion Poisson BCs, no velocity damping
(i) Constant ambient pressure and density (Damp018, 26 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION: Star becomes square-shaped, starts to become unstable after a few dynamical times.

(ii) Isothermal hydrostatic atmosphere (Atm008)
​2d density ​2d density and velocity ​2d density extended range
DESCRIPTION: Star becomes unstable after about 1 dynamical time.

d) Fixed pressure on boundary with rho and veloc extrapolated hydro BCs, Multipole expansion Poisson BCs, no velocity damping
(i) Constant ambient pressure and density (Damp023, 6.6 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity
DESCRIPTION: Star becomes square-shaped, starts to become unstable after a few dynamical times.

e) Fixed pressure and density on boundary with veloc extrapolated hydro BCs, Multipole expansion Poisson BCs, no velocity damping
(i) Constant ambient pressure and density (Damp025, 9 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity
DESCRIPTION: Star becomes square-shaped, starts to become unstable after a few dynamical times.

f) Fixed pressure outside spherical region with rho and veloc extrapolated hydro BCs, Multipole expansion Poisson BCs, no velocity damping
(i) Constant ambient pressure and density (Damp024, 7.7 hrs on comet compute, 576 cores)
​2d density and velocity
DESCRIPTION: Ambient medium becomes unstable and instabilities propagate inward after a few dynamical times, destroying star.

(ii) Isothermal hydrostatic atmosphere (pending on stampede) (Atm010)
​2d density and velocity
DESCRIPTION: Very large velocities at early times at corners of grid.

g) Fixed profile outside sphere of radius 4\times10^{12} cm instead of 5\times10^{12} cm, Multipole expansion Poisson BCs, no velocity damping
(i) Constant ambient pressure and density (Damp026, 26.6 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity
DESCRIPTION: Star becomes square-shaped, starts to become unstable after a few dynamical times.

h) Fixed pressure outside sphere (r=4\times10^{12} cm) with rho and veloc extrapolated hydro BCs, Multipole expansion Poisson BCs, no velocity damping
(i) Constant ambient pressure and density (Damp027, 9.6 hrs on comet compute, 576 cores)
​2d density ​2d density and velocity
DESCRIPTION: Ambient medium becomes unstable and instabilities propagate inward after a few dynamical times, destroying star.

i) Reflecting hydro BCs, Multipole expansion Poisson BCs, no velocity damping
(i) Constant ambient pressure and density (Damp028, 6.9 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity
DESCRIPTION: Star becomes square-shaped, starts to become unstable after a few dynamical times.

(ii) Isothermal hydrostatic atmosphere, no velocity damping (Atm011, 11 hrs on bluehive standard, 120 cores)
​2d density ​2d density and velocity ​2d density extended ​2d density and velocity extended
DESCRIPTION: Star preserves its shape rather well but region exterior to star begins to be unstable after a few dynamical times.

Comparison with (a) on left and (b) on right
(i) Constant ambient pressure and density
​2d density ​2d density and velocity

Comparison with (b) on left and © on right
(i) Constant ambient pressure and density
​2d density ​2d density and velocity

Comparison with (b) on left and (d) on right
(i) Constant ambient pressure and density
​2d density ​2d density and velocity

Comparison with (a) on left and (i) on right
(i) Constant ambient pressure and density
​2d density ​2d density and velocity

Comparison with (b) on left and (i) on right
(i) Constant ambient pressure and density
​2d density ​2d density and velocity

• Fixing the profile on the boundary (b) results in a somewhat more stable star compared with the fiducial case (a).

• Fixing the profile outside a sphere © results in a marginally more stable star compared with case (b).

• Using reflecting hydro BCs (i) gives almost identical results as fixing the profile at the boundary (b).

Conclusions:

• The marginal improvement in going from case (b) to case © probably does not justify the need to artificially fix the hydrodynamical variables within the computation zone. But anyway, we consider both cases when we include damping below.

• Fixing P on a spherical boundary while leaving other variables unconstrained may prevent inflow (Oliger+Sundstrom78, Rudy+Strikwerda80). This should be tried before we go on to longer more computationally intensive runs. Although in case (h) the star retains its spherical morphology for longer, other instabilities are driven due to inflow. It might be worth trying this case with damping.

• Interestingly, reflecting BCs (i) are almost as good as fixing the profile at the boundaries (b). Since the former avoids the extra computation step of resetting the boundary to the initial profile every time step, reflecting hydro BCs are probably preferable to fixing the profile at the boundary.

III) Damping

Motivation:
To improve the stability we now add an artificial damping of the velocity.

Setup:
In ProblemBeforeStep, we convert from conservative to primitive mode and alter the velocity, as discussed last blog post (implementation 3).

Results:

(NOTE THAT VELOCITY VECTORS ARE SCALED 10 TIMES LARGER FOR CASES WITH DAMPING UNLESS STATED OTHERWISE)
a) Fixed profile on boundary hydro BCs, Multipole expansion Poisson BCs, no velocity damping (same as II(b) above)
(i) Constant ambient pressure and density (Damp017, 9 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION: Star becomes square-shaped, starts to become unstable after a few dynamical times.

(ii) Isothermal hydrostatic atmosphere (Atm004)
​2d density ​2d density and velocity
DESCRIPTION: Star becomes unstable after about 1 dynamical time.

b) Fixed profile on boundary hydro BCs, Multipole expansion Poisson BCs, Velocity damping with \tau=1\times10^5 s
(i) Constant ambient pressure and density (Damp021, 9.4 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION: Square-shaped morphology and velocities reduced from case III(a), but still begins to be unstable after a few dynamical times.

c) Fixed profile outside sphere hydro BCs, Multipole expansion Poisson BCs (same as II© above), no velocity damping
(i) Constant ambient pressure and density (Damp018, 26 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION: Star becomes square-shaped, starts to become unstable after a few dynamical times.

(ii) Isothermal hydrostatic atmosphere (Atm008)
​2d density ​2d density and velocity
DESCRIPTION: Ambient medium becomes unstable after about 1 dynamical time.

d) Fixed profile outside sphere hydro BCs, Multipole expansion Poisson BCs, Velocity damping with \tau=1\times10^5 s
(i) Constant ambient pressure and density (Damp022, 26 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION: Square-shaped morphology and velocities reduced from case III(a), but still begins to be unstable after a few dynamical times.

(ii) Isothermal hydrostatic atmosphere (Atm009)
​2d density and velocity ​2d density and velocity (10x smaller arrows) ​2d density and velocity (extended color bar, smaler arrows)
DESCRIPTION: Rather steady and preserves shape rather well at ~1 dynamical time.

e) Reflecting hydro BCs, Multipole expansion Poisson BCs, Velocity damping with \tau=1\times10^5 s
(i) Constant ambient pressure and density (Damp029, 6.9 hrs on bluehive standard 120 cores)
​2d density ​2d density and velocity
DESCRIPTION: Square-shaped morphology and velocities reduced from case III(a), but still begins to be unstable after a few dynamical times.

(ii) Isothermal hydrostatic atmosphere (Atm012, 11.5 hrs on bluehive standard, 120 cores)
​2d density ​2d density and velocity (10x smaller arrows) ​2d density (extended color bar) ​2d density and velocity (extended color bar, 10x smaller arrows)
DESCRIPTION: Preserves shape rather well, but instabilities build up in ambient medium and propagate inward after a few dynamical times.

Comparison with (a) on left and (b) on right
(i) Constant ambient pressure and density
​2d density ​2d density and velocity

Comparison with © on left and (d) on right
(i) Constant ambient pressure and density
​2d density ​2d density and velocity

Comparison with (b) on left and (d) on right
(i) Constant ambient pressure and density
​2d density ​2d density and velocity

• Damping preserves the morphology better and helps the star to remain stable. This is consistent with results from the last blog post with different boundary conditions.

• Fixing the profile at the boundary actually improves stability slightly compared to fixing the profile outside a sphere. (Last plot.)

Conclusions:

• The most successful model for keeping the star stable is III(b) above. Thus III(b) will now be treated as fiducial.

• From past experiments, we know that stability should improve with increased resolution and larger box size. This should allow larger values of \tau to be imployed, as the current value of 1\times10^5 is only about 0.2 dynamical times, smaller than what is used by Ohlmann.

IV) Large box (double box size and number of cells to 512^3, so same physical resolution same)

Motivation:

Setup:

Results:

a) Extrapolated hydro BCs, Multipole expansion Poisson BCs, Velocity damping with \tau=1\times10^5s
(i) Constant ambient pressure and density (Damp013)
​2d density ​2d density and velocity ​2d pressure ​1d density ​1d pressure
DESCRIPTION:

V) AMR with large box

Motivation:

Setup:

Results:

a) Reflecting hydro BCs, Multipole expansion Poisson BCs, Velocity damping with \tau=1\times10^5s, 64^3, maxlevel=4
(i) Constant ambient pressure and density (Damp044) —arrows scaled as in section II without damping
​2d density ​2d density and velocity
DESCRIPTION:

b) Reflecting hydro BCs, Multipole expansion Poisson BCs, No velocity damping, 64^3, maxlevel=4
(ii) Isothermal hydrostatic atmosphere (Atm013)
​2d density ​2d density and velocity
DESCRIPTION: