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