Some recipes for BSSN formulation Some recipes for BSSN formulation - - PowerPoint PPT Presentation

Some recipes for BSSN formulation Some recipes for BSSN formulation

2012 Asia Pacific School/Workshop on Cosmology and Gravitation 2012 Asia Pacific School/Workshop on Cosmology and Gravitation Yukawa Institute for Theoretical Physics, Kyoto, Japan, March 3rd, 2012 Yukawa Institute for Theoretical Physics, Kyoto, Japan, March 3rd, 2012

游輝樟 游輝樟 /Hwei-Jang Yo /Hwei-Jang Yo National Cheng-Kung University, Taiwan National Cheng-Kung University, Taiwan

3+1 Arnowitt-Deser-Misner (ADM) formulation 3+1 Arnowitt-Deser-Misner (ADM) formulation

12 evolution equations & 4 constraints (in vacuum) 12 evolution equations & 4 constraints (in vacuum)

Evolution: d d t i j=−2  Ki j ⇐ d d t ≡ ∂ ∂ t − ℒ 



d d t Ki j= Ri j−2 Ki m K

m j  K Ki j− ∇i ∇ j 

d s

2=− d t 2i j d x i i d t d x j  j d t

31 : M

4

= R× 

3

i j t , x : metric Ki j t , x : extrinsic curvature Constraints: H ≡ R K

2− Ki j K i j≃0

M i≡ ∇ j K

j i− ∇i K ≃0

Baumgarte-Shapiro-Shibata-Nakamura (BSSN) Formulation Baumgarte-Shapiro-Shibata-Nakamura (BSSN) Formulation Features: Features: 1st derivative in time, 2nd derivative in space A conformal decomposition of the metric and the traceless components of the extrinsic curvature. has been shown to be superior to the standard ADM formulation in terms of both accuracy and stability. Strongly hyperbolic with suitable gauges

 ≡ 1 12 ln  ,



i j ≡ e

−4  i j

K ≡ i j K i j ,



Ai j ≡ e−4  K 〈i j〉



i ≡



 j k   j k

i

BSSN variables:

d d t  = −1 6  K d d t  ij = −2   Aij d d t K =   Aij  A

ij1

3 K

2−∇ 2 

d d t  Aij =  K  Aij −2  Aik  A

k je −4   R〈ij 〉−∇ 〈i ∇ j 〉 

∂ ∂ t  

i

= 2   

i jk 

A

jk−2

3  

ij K , j 6 

A

ij , j−2 

A

ij , j

 

jk  i , jk 1

3  

ij  k , jk  j 



i , j −



j  i , j2

3  

i  j , j

Field equations Field equations:

The constraints of H & Mi were used to eliminate in and in .

∂t  

i

d K d t ∂ j  A

i j



R

Constraints Constraints

Hamiltonian constraint: Momentum constraints: Traceless constraint: Unimodular determinant constraint: Gamma constraint:

G

i≡



i− 



j k 



i j k ≃0

H =e

−4  

R−8  ∇

2 −8 

∇

i  

∇i 2 3 K

2− 

Ai j  A

i j≃0

M

i=

∇ j  A

i j6 

A

i j , j−2

3  

i j K , j≃0

A≡  

i j 

Ai j≃0



≡det  i j≃1

Alcubierre & Brügmann [PRD '02]

(1) ( later)

(2) substitute undifferentiated with . Yo, Shapiro & Baumgarte [PRD '02] (1) (2)



z z ⇐



≃1



Az z ⇐ A≃0 ∂t  i  ∂t  i−2 3  Gi  j

, j

= [2 3   

j k 



i j k − 



i]  ℓ , ℓ− 



j  i , j⋯





j k 



i j k



Ai j   A〈i j〉





i



i j   

−1/3 

i j

Modification of the formulation I Modification of the formulation I

∂t  

i=−4

3   

i j K , j⋯=−4

3 [ 

i j K, jK 



i]⋯ ⇐ 



i⇐−



i j , j

Advantage: stabilize the code without the need of the substitute Caveat:

K should be positive, or

a further modification is needed.





i  



j k 



i j k



z z ⇐



≃1



Ay y ⇐ A≃0

Irreducible Decomposition Irreducible Decomposition





i j k=



i j kV i j kU i j k=



i j k3

5 〈 j

i 



ℓ k 〉ℓ−1

5 

i

〈 j 

 k 〉1 3   j k  

i

V

i j k=1

3   j k  

i

⇐





i≡



j k 



i j k

U

i j k=3

5 〈 j

i 



ℓ k 〉 ℓ−1

5 

i

〈 j 

 k 〉 ≃−1 5 

i

〈 j 

 k 〉 ⇐





ℓ iℓ=∂i ln

≃0





i j k=the traceless part of 



i j k=S i j kA i j k

Si j k= i j k=1 3  i j k  j k i k i j Ai j k= i j k−Si j k=2 3  i j k−  j ki

Constraint Application Constraint Application





i j k 



i j k1

3   j kG

i−1

5 

i

〈 jG k 〉−3

5 〈 j

i 



ℓ k 〉 ℓ

III Similarly, ∂i   j k∂i   j k−1 3   j k  

ℓ iℓ1

5  i 〈 j  

ℓ k 〉ℓ3

5  i 〈 jG k 〉 ⇒ 

i∂i 

 j k 

i∂i 

 j k3 5 〈 j G k 〉−1 3   j k 

i 



ℓ iℓ1

5 〈 j  

ℓ k 〉 ℓ

II'

Modification of the formulation II Modification of the formulation II

1 1st

st derivative of the conformal metric

derivative of the conformal metric =2 5 

i∂i 

 j k 

i∂i 

 j k ( jG k )−1 5   j k 

iGi

II

Modification of the formulation III Modification of the formulation III

connection reconstruction connection reconstruction





i j k  



i j k1

3   j kG

i−1

5 

i

〈 j G k 〉−3

5 〈 j

i 



ℓ k 〉ℓ

III

Application of Momentum Constraint Application of Momentum Constraint

∂i  A j k∂i  A j k−1 3   j k Ai− 1 10  i 〈 j Ak 〉−3 5  i 〈 j M k 〉 ⇒ 

i∂i 

A j k 

i∂i 

A j k−3 5 〈 j M k 〉−1 3   j k 

i Ai−1

5 〈 j Ak 〉 ℓ V where Ai≡∂i A=  

j k 

A j k ,i−2  A j k  

j k i≈0

Yoneda & Shinkai ['02]: Decomposition on extrinsic curvature d dt  Ai j d dt  Ai j A  ∇(i M j ) ⇒ d d t  Ai j  d dt  Ai jh f M 〈i, j 〉 IV

I+II+III+IV+V

Summary Summary The modifications focus on the BSSN physical variables The recipes are able to suppress instability efficiently (at least in single Kerr-Schild BH). Need to understand further the behavior of these modifications in the field equations. Need to test these recipes in rotating BHs or BBH.