Zhoujian CaoInstitute of Applied Mathematics, AMSS 2013-10-23

Workshop on Collapsing Objects, Fudan University

Generalized Bondi-Sachs equations for Numerical Relativity

Outline

• Features of numerical relativity code AMSS-NCKU

• Motivation for generalized Bondi-Sachs equations

• Generalized Bondi-Sachs equations for numerical relativity

• Summary

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

2+2:

Characteristic formulation

Formulations implemented

• BSSNOK [Shibata and Nakamura PRD 52, 5428 (1995), Bau

mgarte and Shapiro PRD 59, 024007 (1998)]

• Z4c [Bernuzzi and Hilditch PRD 81, 084003 (2010), Cao and Hil

ditch PRD 85, 124032 (2012)]

• Modified BSSN [Yo, Lin and Cao PRD 86, 064027(2012)]

• Bondi-Sachs [Cao IJMPD 22, 1350042 (2013)]

Mesh refinement

Mesh refinement

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

Mesh refinement

Hilditch, Bernuzzi, Thierfelder, Cao, Tichy and Brugeman (2012)

Code structureMPI + OpenMP + CUDA

Outline

• Features of numerical relativity code AMSS-NCKU

• Motivation for generalized Bondi-Sachs equations

• Generalized Bondi-Sachs equations for numerical relativity

• Summary

BBH models for GW detection

Comparison between our result and calibrated EOB model

t

Cowork with Yi Pan (2012)

BBH models for GW detection

?????

Last problem for BBH model

Simulation efficiency (speed):

• PSAMR

• GPU

• Implicit method [Lau, Lovelace and Pfeiffer PRD 84,

084023]

• Cauchy characteristic matching [Winicour Livi

ng Rev. Relativity 15 (2012)]

Last problem for BBH model

Simulation efficiency:

• PSAMR

• GPU

• Implicit method [Lau, Lovelace and Pfeiffer PRD 84,

084023]

• Cauchy characteristic matching [Winicour Livi

ng Rev. Relativity 15 (2012)]

1. Touch null infinity without extra computational cost

2. Save propagation time

T for Cauchy

T for chara

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

Existing characteristic formalisms

• Null quasi-spherical formalism

S2 of constant u and r should admit standard spherical metric

Existing characteristic formalisms

• Southampton Bondi-Sachs formalism

Existing characteristic formalisms

• Pittsburgh Bondi-Sachs formalism

Relax the form requirement, but essentially r is the luminosity distance parameter

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

Outline

• Features of numerical relativity code AMSS-NCKU

• Motivation for generalized Bondi-Sachs equations

• Generalized Bondi-Sachs equations for numerical relativity

• Summary

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

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

Generalized Bondi-Sachs equations

In order to be a characteristic formalism, we need nested ODE structure, fortunately we have!

Generalized Bondi-Sachs equations

There is no term involved, so for given , it’s ODE

Generalized Bondi-Sachs equations

Generalized Bondi-Sachs equations

There is no term involved, it’s second order ODE system

i,j = 1,2,3

Generalized Bondi-Sachs equations

Generalized Bondi-Sachs equations

There is no term involved, it’s ODE system

Given on

get

get

get

update to

Cowork with Xiaokai He (2013)

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)]

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

Possible application of GBS to CCM

Cauchy Characteristic

Cartesian Spherical

Design equation to control by try and error

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