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 /Hwei-Jang Yo 游輝樟 /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 3  1 : M = R ×  4 3 i d t   d x j d t  2 =− d t 2  i j  d x i  j   d s  i j  t , x  : metric K i j  t , x  : extrinsic curvature 12 evolution equations & 4 constraints (in vacuum) 12 evolution equations & 4 constraints (in vacuum) H ≡ R  K 2 − K i j K i j ≃ 0 Constraints: M i ≡ ∇ j K i − ∇ i K ≃ 0 j ≡ ∂ d d  i j =− 2  K i j ⇐ − ℒ  Evolution:  ∂ t d t d t d K i j =  R i j − 2 K i m K m j  K K i j − ∇ i ∇ j  d t

Baumgarte-Shapiro-Shibata-Nakamura (BSSN) Formulation Baumgarte-Shapiro-Shibata-Nakamura (BSSN) Formulation Features: Features: 1 st derivative in time, 2 nd 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 − 4   i j 1  ≡ 12 ln  ,  i j  ≡ e BSSN variables:  i j K i j , e − 4  K 〈 i j 〉  ≡ ≡ K A i j  j k    i ≡  j k i 

Field equations: Field equations d − 1 d t  = 6  K d − 2    ij = d t  A ij 2  d ij  1    A ij  = 2 −∇ d t K A 3 K − 4   R 〈 ij 〉 −∇ 〈 i ∇ j 〉  d d t    K  A ij − 2  A ik  = k j  e A ij A ∂ ij  , j − 2  ij  , j jk − 2 ij K , j  6  ∂ t  2    jk  i i  =  3   A A A jk  ij  j  j  i  , jk  1 , j  2 , j −  3     i  k , jk   i  i  j 3  , j d K The constraints of H & M i were used to eliminate in  R d t and in . ∂ j  ∂ t  i j i  A

Constraints Constraints Hamiltonian constraint: − 4    2 − 8  i   ∇ i  2 R − 8  2 −  A i j  H = e ∇ ∇ i j ≃ 0 3 K A i j  , j − 2 i =  i j K , j ≃ 0 ∇ j  i j  6  Momentum constraints:  3  M A A i j  A ≡   A i j ≃ 0 Traceless constraint: Unimodular determinant constraint: ≡ det    i j ≃ 1  j k  Gamma constraint: i ≡   i −    i j k ≃ 0 G

Alcubierre & Brügmann [PRD '02] − 1 / 3   A i j    i j     i j  (1) ( later) A 〈 i j 〉 (2) substitute undifferentiated j k     i  with .  i j k Yo, Shapiro & Baumgarte [PRD '02]  z z ⇐ ≃ 1   (1)  ⇐ A ≃ 0 A z z 3  G i  j  i − 2 ∂ t  ∂ t   i  (2) , j = [  2 i ]  j k  j  j k −  ℓ , ℓ −  3     i   i , j ⋯

Modification of the formulation I Modification of the formulation I i =− 4 i j K , j ⋯=− 4 i j K  , j  K  ∂ t  i ]⋯ ⇐      [     i ⇐−   i j , j 3 3 Advantage: stabilize the code without the need of the substitute i   j k      i j k Caveat: K should be positive, or a further modification is needed.  z z ⇐ ≃ 1    ⇐ A ≃ 0 A y y

Irreducible Decomposition Irreducible Decomposition i  j k  3 k 〉 ℓ − 1  k 〉  1  j k =  j k =  〈 j   j k  ℓ  i  i j k  V i j k  U i  i 5  〈 j  5  i  i 3  j k  j k = 1  j k   i  i ⇐  i ≡    i 3  V j k i  j k = 3 k 〉 ℓ − 1  k 〉 ≃− 1 〈 j  〈 j   i ℓ =∂ i ln   ℓ ℓ i 5  〈 j  5  i 5  i  k 〉 ⇐  ≃ 0 U  j k = the traceless part of   i  i j k = S i j k  A i j k   i j k  = 1 S i j k =  3    i j k    j k i    k i j   i j k − S i j k = 2 A i j k =  3    i j k −    j k  i 

Constraint Application Constraint Application i  j k  1 i − 1 〈 j G k 〉 − 3  j k   ℓ  i  i  j k G 5  i 5  〈 j   III  3  k 〉 ℓ Similarly,  j k − 1 i ℓ  1 k 〉 ℓ  3  j k   i 〈 j  ℓ ℓ ∂ i   j k ∂ i     i 〈 j G k 〉 3  5  5  i   j k  3 5  〈 j G k 〉 − 1 i ℓ  1 5  〈 j  ℓ ℓ ⇒  i ∂ i   j k   i ∂ i   j k     II '  3  k 〉 ℓ

Modification of the formulation II Modification of the formulation II 1 st st derivative of the conformal metric derivative of the conformal metric 1  j k   ( j G k ) − 1  i ∂ i   j k  i ∂ i   j k  i G i  II  5  = 2 5

Modification of the formulation III Modification of the formulation III connection reconstruction connection reconstruction i  j k  1 i − 1 〈 j G k 〉 − 3  j k   ℓ  i  i  j k G 5  i 5  〈 j   III  3  k 〉 ℓ

Application of Momentum Constraint Application of Momentum Constraint Yoneda & Shinkai ['02]: d A i j  d A i j  A     ∇ ( i M j ) d t d t d A i j  d   ⇒ A i j  h f  M 〈 i , j 〉  IV  d t d t Decomposition on extrinsic curvature A j k − 1  j k A i − 1  i 〈 j A k 〉 − 3 ∂ i  A j k ∂ i   i 〈 j M k 〉 3  10  5  A j k − 3 5  〈 j M k 〉 − 1 i A i − 1 i ∂ i  i ∂ i  ⇒  A j k   j k  5  〈 j A k 〉 ℓ  V  3  j k  A j k  A j k ,i − 2  A i ≡∂ i A =    i ≈ 0 j k where

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.