1. Gravitational Collapse in Axisymmetry Collaborators: Matthew Choptuik, CIAR/UBC Eric Hircshmann, BYU Steve Liebling, LIU APS Meeting Albuquerque, New Mexico April 20, 2002 Frans Pretorius UBC http://laplace.physics.ubc.ca/People/fransp/

2. 2 Outline Motivation Overview of the physical system Adaptive Mesh Refinement (AMR) in our numerical code Critical phenomena in axisymmetry Conclusion: “near” future extensions

3. 3 Motivation Our immediate goal is to study critical behavior in axisymmetry: –massless, real scalar field –Brill waves –introduce angular momentum via a complex scalar field Long term goals are to explore a wide range axisymmetric phenomena: –head-on black hole collisions –black hole - matter interactions –incorporate a variety of matter models, including fluids and electromagnetism

4. 4 and are the conjugates to and,respectively Geometry: Matter: a minimally-coupled, massless scalar field All variables are functions of Kinematical variables: Dynamical variables: Physical System

5. 5 Adaptive Mesh Refinement Our technique is based upon the Berger & Oliger algorithm –Replace the single mesh with a hierarchy of meshes –Recursive time stepping algorithm Efficient use of resources in both space and time Geared to the solution of hyperbolic-type equations Use a combination of extrapolation and delayed solution for elliptic equations –Dynamical regridding via local truncation error estimates (calculated using a self-shadow hierarchy) –Clustering algorithms: The signature-line method of Berger and Rigoutsos (using a routine written by R. Guenther, M. Huq and D. Choi) Smallest, non-overlapping rectangular bounding boxes

6. 6 2D Critical Collapse example Initial data that is anti-symmetric about z=0: Initial scalar field profile and grid hierarchy (2:1 coarsened in figure)

7. 7 Anti-symmetric SF collapse Scalar field Weak field evolution

8. 8 Anti-symmetric SF collapse Scalar field Near critical evolution

9. 9 AMR grid hierarchy 17(+1), 2:1 refined levels (2:1 coarsened in figure) magnification factor = 1

10. 10 17(+1), 2:1 refined levels (2:1 coarsened in figure) AMR grid hierarchy magnification factor ~ 17

11. 11 17(+1), 2:1 refined levels (2:1 coarsened in figure) AMR grid hierarchy magnification factor ~ 130

12. 12 17(+1), 2:1 refined levels (2:1 coarsened in figure) AMR grid hierarchy magnification factor ~ 330

13. 13

14. 14 Conclusion “Near” future work –More thorough study of scalar field critical parameter space –Improve the robustness of the multigrid solver, to study Brill wave critical phenomena –Include the effects of angular momentum –Incorporate excision into the AMR code –Add additional matter sources, including a complex scalar field and the electromagnetic field