Some recent results (June 2002) 

The results presented here were obtained with a 2D axisymmetric adaptive code, featuring:
For more details see the preliminary draft of my thesis, available here

Anti-symmetric scalar field critical collapse:

The initial data for this simulation is an imploding pulse of scalar field energy, anti-symmetric about z=0. The anti-symmetry prevents any echoing solution from developing at z=0 (at the the center of the implosion) at threshold. Instead, two local self-similar solutions develop off-center. Here is an animation of the scalar field phi in (rho,z) coordinates: phi.mpg (3.9 MB) (smaller version:   phi_small.mpg (2.43MB)). The color and height of the surface in the animation represents the magnitude of the scalar field.

The above solution was tuned to threshold to within 1 part in 10^16. The anti-symmetry is not preserved exactly during evolution due to numerical errors, and hence closer than around 10^-8 to threshold only the left-hand (+z) echoing solution could be tuned to criticality. Around 3 full echoes are observed in the +z solution, though the self-similar nature of the solution prevents one from seeing this in the above animation in cylindrical coordinates. The following image,  phi_ln(r).jpeg, is a snapshot of the last time of the simulation transformed to a radial logarithmic coordinate r and angular coordinate theta, to better demonstrate the self-similar nature of the solution. The transformation is centered about the +z solution, and so the -z solution is severely distorted in the image.

To illustrate the kind of grid-hierarchies produced during evolution, the following series of images shows phi (in rho,z coordinates) as a 2:1 coarsened wireframe mesh, at the initial time: phi_wf_0.jpeg, and at the latest time, with each successive frame zooming into the region of more refinement: phi_tf_z1.jpeg, phi_tf_z2.jpeg, phi_tf_z3.jpeg, phi_tf_z4.jpeg, phi_tf_z5.jpeg, phi_tf_z6.jpeg.
The base grid has a resolution of 65x129, and up to 24 levels of 2:1 refinement were used during evolution.

Critical collapse of an imploding spherical pulse,  with a 'strong' l=2 angular dependence

From the series of scalar field simulations we have run to date, we have found some evidence that there is an additional, 'mildly ' unstable non-spherical (l=2) mode in scalar field critical collapse, that will cause a near critical solution to eventually bifurcate into two, local centers (see the thesis for more details). In the ongoing effort to try to obtain better evidence for this behavior, we simulated an imploding pulse with a significant l=2 angular distribution. Here is an animation of the near critical solution (to within 10^-16) in logarithmic coordinates: phi.mpg (6.5MB) (smaller version phi_small.mpg). Interestingly, most of the non-spherical part of the scalar field gets 'shed' prior to the first echo, and the resulting self-similar solution only has a small l=2 perturbation early on. This perturbation does grow with time, though too slowly to see a bifurcation after the third echo. The following sequence of wireframe images (again 2:1 coarsened and in rho,z coordinates) show phi at the initial time: phi_0.jpeg, and at a later time, where in the last frame the growing asymmetry is evident:  phi_tf_z1.jpeg, phi_tf_z2.jpeg, phi_tf_z3.jpeg, phi_tf_z4.jpeg, phi_tf_z5.jpeg (solid-render of last frame: phi_tf_z5_s.jpeg).
(The base grid has a resolution of 65x129, and up to 26 levels of 2:1 refinement were used during this evolution.)

Note the possible 'bifurcation' of a near critical solution due to the growth of the unstable mode is a different phenomena than the two centers seen in the antisymmetric example above. In fact, the supposition would be that eventually the +z solution in that simulation will also experience the instability and bifurcate --- we see some evidence for this at late times within the simulation: this figure, phi_ln(r).jpeg, shows phi in logarithmic coordinates near the last echo, but with the colormap chosen to highlight the growing asymmetry.

We are still in the process of trying to isolate the unstable mode, in particular to see whether all simulations exhibit the same, unique mode, and to try to estimate its growth rate.

Earlier Results 

Feb. 2002

Near critical solution, from spherically-symmetric initial data:

Scalar field in ln(r,theta) coordinates: phi_ssb_lnr.mpg (4.4MB)

Near critical solution, from initial data anti-symmetric about z=0:

Scalar field in (rho,z) coordinates:
static view: phi_asym_gv.mpg (2.7MB)
zooming-in: phi_asym_zoom.mpg (3.4MB)

May 2001

comparison between AMR and unigrid solutions

example 1:

Psi, AMR solution (64x128 base grid, 1 level of 4:1 refinement, 3 clusters on level 2): psi_amr_0.jpeg
difference in Psi between 64x128 unigrid and 256x512 unigrid solutions: dpsi_uni_0.jpeg
difference in Psi between above AMR solution and 256x512 unigrid solution:  dpsi_amr_0.jpeg

example 2:

AMR solution

Psi:  amr_psi_bbox_0.jpeg,   amr_psi_no_bbox_0.jpeg
Beta_z: amr_bz_bbox_0.jpeg amr_bz_no_bbox_0.jpeg 

comparison with unigrid solutions (region of comparison is that of the level 2 AMR grids):

Psi: amr_psi_comp_0.jpeg 
Beta_z: amr_bz_comp_0.jpeg 

(id files: graxi_ad_id.control , graxi_ad_id.refctl )


weak field run:

scalar field: amr_phi.mpg 
lapse: amr_alpha.mpg 

(id files: graxi_ad_id.control , graxi_ad_id.refctl )