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/
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Gravitational Collapse in Axisymmetry
Collaborators:Matthew Choptuik, CIAR/UBC
Eric Hircshmann, BYUSteve Liebling, LIU
APS MeetingAlbuquerque, New Mexico
April 20, 2002
Frans Pretorius UBC
http://laplace.physics.ubc.ca/People/fransp/
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Outline
• Motivation
• Overview of the physical system
• Adaptive Mesh Refinement (AMR) in our numerical code
• Critical phenomena in axisymmetry
• Conclusion: “near” future extensions
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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
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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
),,( zt
])(
)[(2222
24222
dedtdz
dtddtdsz
),,( z
),,,,(
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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
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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)
z
),,(),,( tztz
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Anti-symmetric SF collapse
Scalar field
Weak field evolution
z
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Anti-symmetric SF collapse
Scalar field
Near critical evolution
z
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AMR grid hierarchy
17(+1), 2:1 refined levels
(2:1 coarsened in figure)
z
magnification factor = 1
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17(+1), 2:1 refined levels
(2:1 coarsened in figure)
z
AMR grid hierarchy
magnification factor ~ 17
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17(+1), 2:1 refined levels
(2:1 coarsened in figure)
z
AMR grid hierarchy
magnification factor ~ 130
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17(+1), 2:1 refined levels
(2:1 coarsened in figure)
z
AMR grid hierarchy
magnification factor ~ 330
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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
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