Adjusted ADM Formulations for Numerical Relativity – Outline

Constraint Propagation Revisited

– Adjusted ADM Formulations for Numerical Relativity –

Hisa-aki Shinkai

真貝寿明 @ 大阪工業大学 情報科学部

Dept of Information Science, Osaka Institute of Technology, Japan shinkai@is.oit.ac.jp

Outline

A search of a formulation for stable numerical evolution

• Adjust ADM formulation with constraints =⇒ Attractor System

• A New Criteria for adjusting rules

• Numerical test with 3D Teukolsky wave evolution =⇒ Longer Stability

Work with Gen Yoneda Math. Sci. Dept., Waseda Univ., Japan

@ 16th JGRG, Niigata, November, 2006

1

Numerical Relativity and “Formulation” Problem

Numerical Relativity

– Necessary for unveiling the nature of strong gravity

–

Gravitational Wave from colliding Black Holes, Neutron Stars, Supernovae, ...

– Relativistic Phenomena like Cosmology, Active Galactic Nuclei, ...

– Mathematical feedbacks to Singularity, Exact Solutions, Chaotic behavior, ...

– Labratory of Gravitational theories, Higher dimensional models, ...

Neutron Stars / Black Holes

Gravitational Waves

LIGO/VIRGO/GEO/TAMA,...

The standard approach :: Arnowitt-Deser-Misner (ADM) formulation (1962)

3+1 decomposition of the spacetime.

Evolve 12 variables (γ_ij, K_ij)

with a choice of gauge condition. surface normal line coordinate constant line

Nⁱ

lapse function, N

shift vector, Nⁱ

t = constant hypersurface

Maxwell eqs. ADM Einstein eq.

constraints div E = 4πρ div B = 0

(3)R + (trK)² − KijK^ij = 2κρH + 2Λ DjK^j_i − DitrK = κJi

evolution eqs.

1

c∂_tE = rot B− 4π c j 1

c∂tB = −rot E

∂_tγ_ij = −2N K_ij +D_jN_i +D_iN_j,

∂_tK_ij = N( ⁽³⁾R_ij + trKK_ij) − 2N K_ilK^l_j − D_iD_jN

+ (D_jN^m)K_mi + (D_iN^m)K_mj +N^mD_mK_ij −N γ_ijΛ

− κα{S_ij + ¹₂γ_ij(ρ_H − trS)}

Best Einstein formulation for long-term stable and accurate simulation?

Many (too many) trials and errors, not yet a systematical understanding.

time

error

Blow up

t=0

Constrained Surface

(satisfies Einstein's constraints)

Best Einstein formulation for long-term stable and accurate simulation?

Many (too many) trials and errors, not yet a systematical understanding.

time

error

Blow up Blow up

ADM

BSSN

Mathematically equivalent formulation, but differ in its stability!

strategy 0: Arnowitt-Deser-Misner formulation

strategy 1: Shibata-Nakamura’s (Baumgarte-Shapiro’s) modifications to the standard ADM strategy 2: Apply a formulation which reveals a hyperbolicity explicitly

strategy 3: Formulate a system which is “asymptotically constrained” against a violation of constraints

Key Fact: By adding constraints in RHS, we can kill error growing modes.

2

Idea of “Adjusted system” and Our Conjecture

Formulate a system which is “asymptotically constrained” against a violation of constraints

“Asymptotically Constrained System”– Constraint Surface as an Attractor

t=0

Constrained Surface

(satisfies Einstein's constraints)

method 1: λ-system (Brodbeck et al, 2000)

• Add aritificial force to reduce the violation of con- straints

• To be guaranteed if we apply the idea to a symmet- ric hyperbolic system.

method 2: Adjusted system (Yoneda HS, 2000, 2001)

• We can control the violation of constraints by ad- justing constraints to EoM.

• Eigenvalue analysis of constraint propagation equa- tions may prodict the violation of error.

• This idea is applicable even if the system is not sym- metric hyperbolic. ⇒

for the ADM/BSSN formulation, too!!

The Idea

General Procedure

1.

prepare a set of evolution eqs.

∂tu^a = f(u^a, ∂bu^a,· · ·)

2.

add constraints in RHS

∂_tu^a = f(u^a, ∂_bu^a,· · ·) +F(C^a, ∂_bC^a,· · ·) 3.

choose appropriate

F(C^a, ∂_bC^a,· · ·)

to make the system stable evolution

How to specify F(C^a, ∂_bC^a,· · ·) ?

4.

prepare constraint propagation eqs.

∂_tC^a = g(C^a, ∂_bC^a,· · ·)

5.

and its adjusted version

∂_tC^a = g(C^a, ∂_bC^a,· · ·) +G(C^a, ∂_bC^a,· · ·) 6.

Fourier transform and evaluate eigenvalues

∂_tCˆ^k = A( ˆC^a)Cˆ^k

Conjecture: Evaluate eigenvalues of (Fourier-transformed) constraint propagation eqs.

If their (1) real part is non-positive, or (2) imaginary part is non-zero, then the system is more stable.

3

Adjusted ADM systems

We adjust the standard ADM system using constraints as:

∂_tγ_ij = −2αK_ij +∇ⁱβ_j +∇^jβ_i, (1) +PijH +Q^kijM^k +p^kij(∇^kH) + q^klij(∇^kM^l), (2)

∂_tK_ij = αR_ij⁽³⁾ +αKK_ij − 2αK_ikK^k_j − ∇ⁱ∇^jα + (∇ⁱβ^k)K_kj + (∇^jβ^k)K_ki +β^k∇^kK_ij,(3) +R_ijH +S^k_ijM^k +r^k_ij(∇^kH) +s^kl_ij(∇^kM^l), (4)

with constraint equations

H := R⁽³⁾ +K² −K_ijK^ij, (5) Mⁱ := ∇^jK^j_i − ∇ⁱK. (6)

We can write the adjusted constraint propagation equations as

∂_tH = (original terms)+ H₁^mn[(2)] +H₂^imn∂_i[(2)] +H₃^ijmn∂_i∂_j[(2)] +H₄^mn[(4)], (7)

∂_tMⁱ = (original terms)+ M_1i^mn[(2)] +M_2i^jmn∂_j[(2)] +M_3i^mn[(4)] + M_4i^jmn∂_j[(4)]. (8)

The constraint propagation equations of the original ADM equation:

• Expression using H and Mⁱ (1)

∂_tH = β^j(∂_jH) + 2αKH − 2αγ^ij(∂_iM^j) +α(∂_lγ_mk)(2γ^mlγ^kj − γ^mkγ^lj)M^j −4γ^ij(∂_jα)Mⁱ,

∂_tMⁱ = −(1/2)α(∂_iH) − (∂_iα)H +β^j(∂_jMⁱ) +αKMⁱ − β^kγ^jl(∂_iγ_lk)M^j + (∂_iβ_k)γ^kjM^j.

• Expression using H and Mⁱ (2)

∂_tH = β^l∂_lH+ 2αKH − 2αγ^−1/2∂_l(√γM^l) −4(∂_lα)M^l

= β^l∇^lH+ 2αKH − 2α(∇^lM^l) − 4(∇^lα)M^l,

∂_tMⁱ = −(1/2)α(∂_iH) − (∂_iα)H +β^l∇^lMⁱ +αKMⁱ + (∇ⁱβ_l)M^l

= −(1/2)α(∇ⁱH) − (∇ⁱα)H+ β^l∇^lMⁱ + αKMⁱ + (∇ⁱβ_l)M^l,

• Expression using H and Mⁱ (3): by using Lie derivatives along αn^µ,

£_αnµH = +2αKH − 2αγ^−1/2∂_l(√γM^l) − 4(∂_lα)M^l,

£_αnµMⁱ = −(1/2)α(∂iH) − (∂iα)H +αKMⁱ.

• Expression using γ_ij and K_ij

∂_tH = H₁^mn(∂_tγ_mn) +H₂^imn∂_i(∂_tγ_mn) + H₃^ijmn∂_i∂_j(∂_tγ_mn) + H₄^mn(∂_tK_mn),

∂tMⁱ = M1imn(∂tγmn) + M2ijmn∂j(∂tγmn) + M3imn(∂tKmn) + M4ijmn∂j(∂tKmn), where

H₁^mn := −2R^(3)mn − Γ^p_kjΓ^k_piγ^miγ^nj + Γ^mΓⁿ

+γ^ijγ^np(∂_iγ^mk)(∂_jγ_kp) − γ^mpγⁿⁱ(∂_iγ^kj)(∂_jγ_kp) − 2KK^mn + 2Kⁿ_jK^mj, H₂^imn := −2γ^miΓⁿ − (3/2)γ^ij(∂jγ^mn) + γ^mj(∂jγⁱⁿ) +γ^mnΓⁱ,

H₃^ijmn := −γ^ijγ^mn +γⁱⁿγ^mj, H₄^mn := 2(Kγ^mn − K^mn),

M_1i^mn := γ^nj(∂_iK^m_j) − γ^mj(∂_jKⁿ_i) + (1/2)(∂_jγ^mn)K^j_i + ΓⁿK^m_i, M_2i^jmn := −γ^mjKⁿ_i + (1/2)γ^mnK^j_i + (1/2)K^mnδ_i^j,

M_3i^mn := −δ_iⁿΓ^m − (1/2)(∂_iγ^mn), M4ijmn := γ^mjδⁿ_i −γ^mnδ_i^j,

where we expressed Γ^m = Γ^m_ijγ^ij.

4

Additional Idea (NEW)

In order to avoid blow-up in the last stage, we prohibid the adjustments which simply produce self-growing terms (C

²

) in constraint propagation, ∂

_t

C .

• If RHS of the constraint propagation accidentally includes C² terms,

∂_tC = −aC +bC² the solution will blow-up as

C = −aC0exp(−at)

−a +bC₀ − bC₀exp(−at)

In the ADM system, we have not to put too much confidence for the adjustments using p, q, P, Q- terms for the ADM formulation.

∂_tγ_ij = −2αK_ij +∇ⁱβ_j +∇^jβ_i

+P_ijH+ Q^k_ijM^k + p^k_ij(∇^kH) +q^kl_ij(∇^kM^l),

∂_tK_ij = αR⁽³⁾_ij + αKK_ij −2αK_ikK^k_j − ∇ⁱ∇^jα + (∇ⁱβ^k)K_kj + (∇^jβ^k)K_ki +β^k∇^kK_ij +R_ijH + S^k_ijM^k + r^k_ij(∇^kH) + s^kl_ij(∇^kM^l),

5

Numerical Test

Comparisons of Adjusted ADM systems (linear wave)

Original GR code based on Cactus framework.

1 0 5

1 0 4

1 0 3

1 0 2

1 0 1 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

A D M (st a n d a rd )

si m p

li i f e d Dte w iel e r k= 0 .0 1

si m p

li i f e d Dte w iel e r k= 0 .0 1 , n o c + 0 .0 1

si m p

li i f e d Dte w iel e r k= 0 .0 1 , n o c + 0 .0 5

error(normofHamiltonianconxtraint) tim e

Violation of Hamiltonian constraints versus time:

Adjusted ADM systems applied for Teukolsky wave initial data evolution with harmonic slicing, and with periodic boundary condition. Cactus/GR code was used. Grid = 24³, ∆x = 0.25, iterative Crank- Nicholson method.

=⇒ Newly added term works effectively. 10%

longer evolution is available, but not yet perfect...

(to be continued)

∂_tγ_ij = −2αK_ij +∇ⁱβ_j +∇^jβ_i − κ₁αγ_ijH

∂_tK_ij = αR_ij⁽³⁾ +αKK_ij − 2αK_ikK^k_j − ∇ⁱ∇^jα + (∇ⁱβ^k)K_kj + (∇^jβ^k)K_ki +β^k∇^kK_ij +κ2αγijγ^kl∂kM^l

Comparisons of Adjusted ADM systems (linear wave)

Original GR code based on Cactus framework.

1 0; 6

1 0; 5

1 0; 4

1 0; 3

1 0; 2

1 0; 1 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0

F G H IJ KL M N L ON P

G Q KR Q STQ O UV W XW Y

G Q KR Q STQ O UV W XW Y Z M [ \ UV W XW Y

G Q KR Q STQ O UV W XW Y Z M [ \ UV W XW ]

error(normofHamiltonianconxtraint) ti m e

Violation of Hamiltonian constraints versus time:

Adjusted ADM systems applied for Teukolsky wave initial data evolution with harmonic slicing, and with periodic boundary condition. Cactus/GR code was used. Grid = 24³, ∆x = 0.25, iterative Crank- Nicholson method.

=⇒ Newly added term works effectively. 10%

longer evolution is available, but not yet perfect...

(to be continued)

∂_tγ_ij = −2αK_ij +∇ⁱβ_j +∇^jβ_i − κ₁α³γ_ijH

∂_tK_ij = αR_ij⁽³⁾ +αKK_ij − 2αK_ikK^k_j − ∇ⁱ∇^jα + (∇ⁱβ^k)K_kj + (∇^jβ^k)K_ki +β^k∇^kK_ij +κ1α³(Kij − (1/3)Kγij)H +κ2αγijγ^kl ∂kM^l

+κ₁α²[3(∂_(iα)δ_j^k₎ − (∂_lα)γ_ijγ^kl]M^k +κ₁α³[δ_(i^kδ_j)^l −(1/3)γ_ijγ^ki](∇^kM^l)