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A New Code for Axisymmetric Numerical Relativity

A New Code for Axisymmetric Numerical Relativity. Eric Hircshmann, BYU Steve Liebling, LIU Frans Pretorius, UBC Matthew Choptuik CIAR/UBC. Black Holes III Kananaskis, Alberta May 22, 2001. Outline. Motivation Previous Work (other axisymmetric codes) Formalism & equations of motion

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A New Code for Axisymmetric Numerical Relativity

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  1. A New Code for Axisymmetric Numerical Relativity Eric Hircshmann, BYU Steve Liebling, LIU Frans Pretorius, UBC Matthew Choptuik CIAR/UBC Black Holes III Kananaskis, Alberta May 22, 2001

  2. 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

  3. 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!

  4. 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

  5. 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”)

  6. Development of Techniques & Algorithms for General Use • Coordinate choices (lapse and shift) • Black hole excision techniques • Adaptive mesh refinement (AMR) algorithms

  7. 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

  8. 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

  9. 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

  10. 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)

  11. 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

  12. 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)

  13. 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

  14. Geometry • all functions of

  15. Geometry • Coordinate conditions • Diagonal 2-metric • Maximal slicing • Kinematical variables: • Dynamical variables: Conjugate to

  16. Matter • Single minimally coupled massless scalar field, • Also introduce conjugate variable

  17. Evolution Scheme • Evolution equations for • “Constraint” equations for • Also have evolution equation for which is used at times • Compute and monitor ADM mass

  18. Regularity Conditions • As , all functions either go as or • Regularity EXPLICITLY enforced

  19. 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

  20. Initial Data • Freely specify evolved quantities • Solve “constraints” for

  21. Numerical Approach • Use uniform grid in • Grid includes • Use finite-difference formulae (mostly centred difference approximations)

  22. 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

  23. 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

  24. Kreiss-Oliger Dissipation: Example • Consider the simple “advection” equation • Finite difference via and

  25. Kreiss-Oliger Dissipation: Example • Add “Kreiss-Oliger” dissipation via • Where and

  26. Effect of Dissipation65 x 129 Grid

  27. Effect of Dissipation65 x 129 Grid

  28. Effect of Dissipation65 x 129 Grid

  29. Effect of Dissipation129 x 257 grid

  30. Effect of Dissipation129 x 257 grid

  31. Effect of Dissipation129 x 257 grid

  32. Effect of Dissipation127 x 259 grid

  33. Collapse of Oblate and Prolate Scalar Pulses

  34. Collapse of Oblate and Prolate Scalar Pulses

  35. Collapse of Weak Brill Waves

  36. Collapse of Weak Brill Waves

  37. Collapse of Asymmetric Scalar Pulses

  38. Collapse of Asymmetric Scalar Pulses

  39. Convergence

  40. 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

  41. Close Merger of Two Scalar Pulses with Excision

  42. Asymmetric Scalar Collapse with Excision

  43. Asymmetric Scalar Collapse with Excision

  44. Boosted Merger of Two Scalar Pulses with Excision

  45. Boosted Merger of Two Scalar Pulses with Excision

  46. Boosted Merger of Two Scalar Pulses with Excision

  47. “Waveform Extraction”

  48. Black Hole & Brill Wave with Excision

  49. Black Hole & Brill Wave with Excision

  50. Black Hole & Brill Wave with Excision

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