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

7. 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. 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. 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. 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. 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. 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. 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. 14 Geometry all functions of

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

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

17. 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. 18 Regularity Conditions As, all functions either go as or Regularity EXPLICITLY enforced

19. 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. 20 Initial Data Freely specify evolved quantities Solve “constraints” for

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

22. 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. 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. 24 Kreiss-Oliger Dissipation: Example Consider the simple “advection” equation Finite difference via and

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

26. 26 Effect of Dissipation 65 x 129 Grid

27. 27 Effect of Dissipation 65 x 129 Grid

28. 28 Effect of Dissipation 65 x 129 Grid

29. 29 Effect of Dissipation 129 x 257 grid

30. 30 Effect of Dissipation 129 x 257 grid

31. 31 Effect of Dissipation 129 x 257 grid

32. 32 Effect of Dissipation 127 x 259 grid

33. 33 Collapse of Oblate and Prolate Scalar Pulses

34. 34 Collapse of Oblate and Prolate Scalar Pulses

35. 35 Collapse of Weak Brill Waves

36. 36 Collapse of Weak Brill Waves

37. 37 Collapse of Asymmetric Scalar Pulses

38. 38 Collapse of Asymmetric Scalar Pulses

39. 39 Convergence

40. 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. 41 Close Merger of Two Scalar Pulses with Excision

42. 42 Asymmetric Scalar Collapse with Excision

43. 43 Asymmetric Scalar Collapse with Excision

44. 44 Boosted Merger of Two Scalar Pulses with Excision

45. 45 Boosted Merger of Two Scalar Pulses with Excision

46. 46 Boosted Merger of Two Scalar Pulses with Excision

47. 47 “Waveform Extraction”

48. 48 Black Hole & Brill Wave with Excision

49. 49 Black Hole & Brill Wave with Excision

50. 50 Black Hole & Brill Wave with Excision

51. 51 Highly Prolate Scalar Collapse With Excision

52. 52 Highly Prolate Scalar Collapse With Excision

53. 53 Highly Prolate Scalar Collapse With Excision

54. 54 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

55. 55 AMR Results: Initial Data 3 Levels of 4:1 Refinement

56. 56 AMR Results: Initial Data 3 Levels of 4:1 Refinement

57. 57 AMR Results: Initial Data 3 Levels of 4:1 Refinement

58. 58 AMR Results: Intermediate Strength Asymmetric Pulses (5 Levels of 2:1 Refinement)

59. 59 AMR Results: Intermediate Strength Symmetric Pulses

60. 60 AMR Results: Intermediate Strength Symmetric Pulses

61. 61 AMR Results: Intermediate Strength Asymmetric Pulses

62. 62 AMR Results: Intermediate Strength Asymmetric Pulses

63. 63 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

64. 64 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