a new code for axisymmetric numerical relativity eric hircshmann, byu steve liebling, liu frans...
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A New Code for Axisymmetric Numerical Relativity
Eric Hircshmann, BYUSteve Liebling, LIU
Frans Pretorius, UBCMatthew Choptuik CIAR/UBC
Black Holes IIIKananaskis, Alberta
May 22, 2001
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Outline
• Motivation• Previous Work (other axisymmetric codes)• Formalism & equations of motion• Numerical considerations• Early results• Black hole excision results• Adaptive mesh refinement results• Future work
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Motivation
• Construct accurate, robust code for axisymmetric calculations in GR
• Full 3D calculations still require more computer resources than typically available (especially in Canada!)
• Interesting calculations to be done!
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Long Term Goals
• Critical Phenomena
– Test non-linear stability of known spherical solutions
– Look for new solutions with new matter sources
– Study effects of rotation
– Repeat Abrahams & Evans gravitational-wave collapse calculations with higher resolution
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Long Term Goals
• Cosmic Censorship
– Reexamine Shapiro & Teukolsky computations suggesting naked singularity formation in highly prolate collapse but using matter with better convergence properties
• ??? (“Expect the Unexpected”)
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Development of Techniques & Algorithms for General Use
• Coordinate choices (lapse and shift)
• Black hole excision techniques
• Adaptive mesh refinement (AMR) algorithms
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Previous Work (“Space + Time” Approaches)
• NCSA / Wash U / Potsdam … (Smarr & Eppley (1978), Hobill, Seidel, Bernstein, Brandt …)
– Focused on head-on black hole collisions using “boundary conforming” (Cadez) coordinates
– Culminates in work by Brandt & Anninos (98-99); head-on collisions of different-massed black holes, estimation of recoil due to gravity-wave emission
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Previous Work
• Nakamura & collaborators (early 80’s)
– Rotating collapse of perfect fluid using (2+1)+1 approach
• Stark & Piran (mid 80’s)
– Rotating collapse of perfect fluid, relatively accurate determination of emitted gravitational wave-forms
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Previous Work
• Cornell Group (Shapiro,Teukolsky, Abrahams, Cook …)
– Studied variety of problems in late 80’s through early 90’s using non-interacting particles as matter source
• FOUND EVIDENCE FOR NAKED SINGULARITY FORMATION IN SUFFICIENTLY PROLATE COLLAPSE
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Previous Work
• Evans, (84-), Abrahams and Evans (-93)
– Began as code for general relativistic hydrodynamics
– Later specialized to vacuum collapse (Brill waves)
• STUDIED CRITICAL COLLAPSE OF GRAVITATIONAL WAVES; FOUND EVIDENCE FOR SCALING & UNIVERSALITY (93)
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Problems With Axisymmetry
• Most codes used polar/spherical coordinates
• Severe difficulties with regularity at coordinate singularities: , but especially -axis
• Long-time evolutions difficult due to resulting instabilities
• MAJOR MOTIVATION FOR SUSPENSION OF 2D STUDIES IN MID-90’s
0r z
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Formalism
• Adopt a (2+1)+1 decomposition; dimensional reduction --- divide out the action of the Killing vector (Geroch)
• Gravitational degrees of freedom in 2+1 space
– Scalar:
– Twist vector: (ONE dynamical degree of freedom)
2es
aw
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Formalism
• Have not incorporated rotation yet (no twist vector); in this case easy to relate (2+1)+1 equations to “usual” 3+1 form
• Adopt cylindrical coordinates
• No dependence of any quantities on
),,,( zt
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Geometry
• all functions of ),,( zt
])(
)[(2222
24222
dedtdz
dtddtdsz
),,,,( z
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Geometry
• Coordinate conditions
– Diagonal 2-metric
– Maximal slicing
• Kinematical variables:
• Dynamical variables:
0)3( K
),,( z
Conjugate to
),,(
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Matter
• Single minimally coupled massless scalar field,
• Also introduce conjugate variable
),,( zt
0
gT 2
),,( zt
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Evolution Scheme
• Evolution equations for
• “Constraint” equations for
• Also have evolution equation for which is used at times
• Compute and monitor ADM mass
),,,(
),,,( z
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Regularity Conditions
• As , all functions either go as
or
• Regularity EXPLICITLY enforced
0
)(),(),(),,( 42
20 Oztfztfztf
)(),(),(),,( 53
31 Oztgztgztg
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Boundary Conditions
• Numerical domain is FINITE
• Impose naïve outgoing radiation conditions on evolved variables,
• Conditions based on asymptotic flatness and leading order behaviour used for “constrained” variables
222,, 0)()( zrfrfr rt
),,( ztf
r
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Initial Data
• Freely specify evolved quantities
• Solve “constraints” for
),,,(
),,,( z
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Numerical Approach
• Use uniform grid in
• Grid includes
• Use finite-difference formulae (mostly centred difference approximations)
),,( zt
2/1 hthz
)( 2hO
0
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Numerical Approach
• Use “iterative Crank-Nicholson” to update evolved variables
• Use multi-grid to solve coupled elliptic equations for , based on point-wise simultaneous relaxation of all four variables
• Still have some problems with multi-grid in strong Brill collapse; using evolution equation for helps
),,,( z
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Dissipation
• Add explicit dissipation of “Kreiss-Oliger” form to differenced evolution equations
• Scheme remains (second order), but high-frequency components are effectively damped
• CRUCIAL for controlling instabilities, particularly along -axis
)( 2hO
z
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Kreiss-Oliger Dissipation: Example
• Consider the simple “advection” equation
• Finite difference via and
x
uu
t
uu nj
nj
nj
nj
2211
11
uu xt
),( xjtnuunj
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Kreiss-Oliger Dissipation: Example
• Add “Kreiss-Oliger” dissipation via
• Where and
421124 464
x
uuuuuuD
nj
nj
nj
nj
njn
j
1441
161
n
jnj uDxu
10
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Effect of Dissipation65 x 129 Grid
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Effect of Dissipation65 x 129 Grid
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Effect of Dissipation65 x 129 Grid
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Effect of Dissipation129 x 257 grid
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Effect of Dissipation129 x 257 grid
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Effect of Dissipation129 x 257 grid
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Effect of Dissipation127 x 259 grid
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Collapse of Oblate and Prolate Scalar Pulses
),,( zt
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Collapse of Oblate and Prolate Scalar Pulses
),,( zt
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Collapse of Weak Brill Waves
),,( zt
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Collapse of Weak Brill Waves
),,( zt
37
Collapse of Asymmetric Scalar Pulses
),,( zt
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Collapse of Asymmetric Scalar Pulses
),,( zt
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Convergence
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Black Hole Excision
• To avoid singularity within black hole, exclude interior of hole from computational domain (Unruh)
• Operationally, track some surface(s) interior to apparent horizon(s)
• Currently fix excision surface by scanning level contours of a priori specified function and choosing surface on which outgoing divergence of null rays is sufficiently negative
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Close Merger of Two Scalar Pulses with Excision
),,( zt
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Asymmetric Scalar Collapse with Excision
),,( zt
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Asymmetric Scalar Collapse with Excision
),,( zt
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Boosted Merger of Two Scalar Pulses with Excision
),,( zt
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Boosted Merger of Two Scalar Pulses with Excision
),,( zt
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Boosted Merger of Two Scalar Pulses with Excision
),,( zt
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“Waveform Extraction”
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Black Hole & Brill Wave with Excision
),,( zt
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Black Hole & Brill Wave with Excision
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2 ),,( ztr
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Black Hole & Brill Wave with Excision
2
04
~),,~(
tdztrt
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Highly Prolate Scalar Collapse With Excision
),,( zt
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Highly Prolate Scalar Collapse With Excision
),,( zt
53
Highly Prolate Scalar Collapse With Excision
),,(4 ztr
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Adaptive Mesh Refinement
• After stability, adequate resolution is THE KEY to successful solution of time-dependent PDEs
• Problems of interest to us exhibit enormous dynamical range; adaptive mesh refinement essential (arguably more important than parallelization)
• Current approach based on Berger & Oliger algorithm (84), follows work done in 1d, and modules from Binary BH Grand Challenge
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AMR Results: Initial Data3 Levels of 4:1 Refinement
),,0( z
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AMR Results: Initial Data3 Levels of 4:1 Refinement
),,0( zz
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AMR Results: Initial Data3 Levels of 4:1 Refinement
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AMR Results: Intermediate Strength Asymmetric Pulses(5 Levels of 2:1 Refinement)
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AMR Results: Intermediate Strength Symmetric Pulses
),,( zt
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AMR Results: Intermediate Strength Symmetric Pulses
),,( ztr
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AMR Results: Intermediate Strength Asymmetric Pulses
),,( ztr
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AMR Results: Intermediate Strength Asymmetric Pulses
),,( ztr
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Future Work
• AMR current focus; needed for virtually all long-term goals of project
• Once fully implemented, will investigate various issues in critical collapse, highly prolate collapse etc., using scalar fields and gravitational waves
• Plan to add Maxwell field shortly, collapse of strong EM fields essentially unstudied
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Future Work
• Will continue to work on excision techniques and head-on collisions of black holes, and other compact objects (e.g. boson stars)
• Will add rotation, study effect on critical collapse of complex scalar fields, EM field
• Longer term---will incorporate relativistic hydrodynamics