AMR/RT Meeting, April 9th, 2008


Table of Contents
  1. The Status at Our Last Meeting
  2. Last Meeting's "Future Work"
  3. Current Status
  4. Current "Future Work"


The Status at Our Last Meeting

The main motivation of the project is the speedup associated with Adaptive Mesh Refinement (AMR) over Fixed Grid codes. This speedup grants us access to what would otherwise be prohibitively expensive calculations of uninvestigated physics. The relevant example is the three dimensional, multiple-Fourier-mode, ablatively supported Rayleigh-Taylor instability.

AstroBEAR benefits from both AMR and parallelization. The graph below illustrates these gains over a fixed grid, single-processor code:


Black: Fixed grid, 256x1024. Red: AMR, 64x256+2. Blue: Region-criteria AMR.
  1. At 16 processors we have a speedup of ~3.5x. (Note: newer versions of astroBEAR have improved this performance gain.)
  2. The single-processor AMR is faster yet by a factor of 1.5, being 5x faster than the fixed grid. Parallel AMR displays similar gains to the parallel fixed grid.

We found that doing AMR in the usual manner -- refining spatial discontinuities in the fluid variables -- leads to spurious features at the interface. If you instead refine a region around the interface, you maintain much of the AMR benefit, without spurious effects. The problem that generated the above plot received a 5-fold speedup from AMR.


Region-criteria AMR and Fixed Grid Return the Same (Correct) Growth

The following graph illustrates that fixed grid (blue) and AMR (red) evolve identically, into the nonlinear growth regime.


Blue: fixed grid, 256x1024. Red: AMR, 64x256+2. A plot of the calculated bubble growth as a function of time shows the two methods agree (up until the analysis itself breaks down).

A fit to the growth rate returns a value of γ with uncertainty of a few percent which brackets the analytic value for this setup.

γanalytic = (ATkg)1/2 = (0.9048 (2&pi/10μm) (100μm/ns²))1/2
≈ 7.54 ns-1
and     γsim = 7.1±0.7 ns-1 ,

where AT = (ρHL) / (ρHL) is the Atwood number.


Red line: Fit to the data. The red dashed line denotes bubble height of 0.1 λ.

The Linear System Solver Functioned As Expected

The newly-created linear system solver was applied to the equation,

ρcv∂T/∂t = ∇⋅(K ∇T)

with the coefficient K being a constant, this reduces to

∂T/∂t = α ∇2T

where α = K/ρcv is the thermal diffusivity. The linear solver performed in this case,

K = 5e-2 K = 0. No Linear Solve
Left: heat diffusion. Right and Center: setting the coefficient to zero results in the same answer as turning off the solver.

Heat conduction with an exponent above 0.3 (K∝T0.3) was found to be too extreme for the solver. (Recall that an exponent of 2.5 was ultimately desired.)


2D Multimode, Classic RT Growth Rate was Confirmed

Instead of perturbing the unstable interface with a single wavemode, one can perturb with a spectrum of modes. This "multimode" seeding will affect the growth and the morphology of the instability.


2D multimode, classic RT. Schlieren (density gradients) image.

The above is an region-AMR simulation with 4 levels total (256x1024 maximum resolution). Mode numbers n=1, 2, 4, 8, 16, 32 corresponding to cells-per-wavelength of 256, 128, 64, 32, 16, 8 were seeded. The perturbation was in velocity; the only difference between this simulation and our previous single-mode simulations was the multimode seed (all other parameters were the same).

Theory suggests the value of the bubble height with respect to time will depend on a bubble growth constant αb,

hb(t) = αb AT g t2

The value of αb is expected to be

αb = 0.05–0.06  for 2D
αb ≈ 0.03  for 3D
We found agreement with the bubble growth constant's value.


A fit to the bubble height early in time returns a value of αb=0.05±0.003

3D Multimode, Classic RT Initialization was Shown to be Consistent with Published Work

Published studies of the 3D multimode RT typically perturb the interface itself rather than seed a velocity perturbation. The wave numbers were chosen with random orientation and a magnitude proportional to k-2. (additional details)

Below is a cross-cut in the x-z plane ("vertical" is in the y-direction) at the interface. The wavemodes to be seeded is user-controlled; below are two examples.

Left: Modes 2-32 seeded. Right: modes 32-64 seeded.

For comparison, below is a published figure from Ramaprabhu et al. 2005. Note that the figure on the left does not depict the same initialization as the above left figure (though it is straightforward to produce such a corresponding image).


Left: modes 16-32 seeded. Right: modes 32-64 seeded.

At the time, difficulties were encountered getting the simulation to run.