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Zhoujian Cao Institute of Applied Mathematics, AMSS 2013-10-23

Generalized Bondi-Sachs equations for Numerical Relativity. Zhoujian Cao Institute of Applied Mathematics, AMSS 2013-10-23. Workshop on Collapsing Objects, Fudan University. Outline. Features of numerical relativity code AMSS-NCKU

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Zhoujian Cao Institute of Applied Mathematics, AMSS 2013-10-23

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

  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

  4. Characteristic formulation 2+2:

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

  6. Mesh refinement

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

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

  9. Code structure MPI + OpenMP + CUDA

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

  11. BBH models for GW detection

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

  13. BBH models for GW detection ?????

  14. 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)]

  15. 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)]

  16. 1. Touch null infinity without extra computational cost

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

  18. 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

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

  20. Existing characteristic formalisms • Southampton Bondi-Sachs formalism

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

  22. 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

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

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

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

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

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

  28. Generalized Bondi-Sachs equations

  29. Generalized Bondi-Sachs equations i,j = 1,2,3 There is no term involved, it’s second order ODEsystem

  30. Generalized Bondi-Sachs equations

  31. Generalized Bondi-Sachs equations There is no term involved, it’s ODE system

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

  33. Given on • Nested ODE structure • 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)] update to

  34. Gauge variable • There is no equation to control • 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

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

  36. 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

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