1. Zhoujian Cao Institute of Applied Mathematics, AMSS 2013-10-23 Workshop on Collapsing Objects, Fudan University Generalized Bondi-Sachs equations for Numerical Relativity

2. Outline Features of numerical relativity code AMSS-NCKU Motivation for generalized Bondi-Sachs equations Generalized Bondi-Sachs equations for numerical relativity Summary

3. NR code AMSS-NCKU Developers include: Shan Bai (AMSS), Zhoujian Cao (AMSS), Zhihui Du (THU), Chun-Yu Lin (NCKU), Quan Yang (THU), Hwei-Jang Yo (NCKU), Jui-Ping Yu (NCKU) 2007-now

5. 2+2: Characteristic formulation

6. Formulations implemented BSSNOK [ Shibata and Nakamura PRD 52, 5428 (1995), Baumgarte and Shapiro PRD 59, 024007 (1998) ] Z4c [ Bernuzzi and Hilditch PRD 81, 084003 (2010), Cao and Hilditch PRD 85, 124032 (2012) ] Modified BSSN [ Yo, Lin and Cao PRD 86, 064027(2012) ] Bondi-Sachs [ Cao IJMPD 22, 1350042 (2013) ]

7. Mesh refinement

8. Parallel structured mesh refinement (PSAMR), co work with Brandt, Du and Loffler, 2013

9. Mesh refinement Hilditch, Bernuzzi, Thierfelder, Cao, Tichy and Brugeman (2012)

10. Code structure MPI + OpenMP + CUDA

11. Outline Features of numerical relativity code AMSS-NCKU Motivation for generalized Bondi-Sachs equations Generalized Bondi-Sachs equations for numerical relativity Summary

12. BBH models for GW detection

13. Comparison between our result and calibrated EOB model t Cowork with Yi Pan (2012)

14. BBH models for GW detection ?????

15. Last problem for BBH model Simulation efficiency (speed): PSAMR GPU Implicit method [ Lau, Lovelace and Pfeiffer PRD 84, 084023 ] Cauchy characteristic matching [ Winicour Living Rev. Relativity 15 (2012) ]

16. Last problem for BBH model Simulation efficiency: PSAMR GPU Implicit method [ Lau, Lovelace and Pfeiffer PRD 84, 084023 ] Cauchy characteristic matching [ Winicour Living Rev. Relativity 15 (2012) ]

17. 1. Touch null infinity without extra computational cost

18. 2. Save propagation time T for Cauchy T for chara

19. Cauchy-Characteristic matching (CCM) Many works have been contributed to CCM [Pittsburgh, Southampton 1990’s] But hard to combine! difficulty 1. different evolution scheme difficulty 2. different gauge condition

20. Existing characteristic formalisms Null quasi-spherical formalism S2 of constant u and r should admit standard spherical metric

21. Existing characteristic formalisms Southampton Bondi-Sachs formalism

22. Existing characteristic formalisms Pittsburgh Bondi-Sachs formalism Relax the form requirement, but essentially r is the luminosity distance parameter

23. Existing characteristic formalisms Affine Bondi-Sachs formalism In contrast to luminosity parameter, affine parameter can be matched to any single layer of coordinate cylinder r

24. Outline Features of numerical relativity code AMSS-NCKU Motivation for generalized Bondi-Sachs equations Generalized Bondi-Sachs equations for numerical relativity Summary

25. Generalized Bondi-Sachs formalism Requirements: 1. is null 2. is hypersurface forming In contrast to the existing Bondi-Sachs formalism, the parameterization of r is totally free A,B = 2,3

26. guarantees that we can use main equations only to do free evolution

27. Generalized Bondi-Sachs equations In order to be a characteristic formalism, we need nested ODE structure, fortunately we have!

28. Generalized Bondi-Sachs equations There is no term involved, so for given, it’s ODE

29. Generalized Bondi-Sachs equations

30. There is no term involved, it’s second order ODE system i,j = 1,2,3

31. Generalized Bondi-Sachs equations

32. There is no term involved, it’s ODE system

33. Given on get update to Cowork with Xiaokai He (2013)

34. Given on update to 1.Nested ODE structure 2.Facilitate us to use MoL which makes us to evolve Cauchy part and characteristic part with the same numerical scheme [Cao, IJMPD 22, 135042 (2013)]

35. Gauge variable 1.There is no equation to control 2. is related to parameterization of r is a gauge freedom, it is possible to use this freedom to relate the gauge used in inner Cauchy part for CCM

36. Possible application of GBS to CCM Cauchy Characteristic Cartesian Spherical Design equation to control by try and error

37. Summary Feature of AMSS-NCKU code Efficiency problem in BBH model CCM can improve efficiency, but the existing characteristic formalisms face difficulties of different numerical scheme and gauge to Cauchy part Generalized BS formalism may help to solve these difficulties through the introduction of gauge freedom