Controlling Constraint Violations in General Relativity

Controlling Constraint Violations in General Relativity

— Asymptotically Constrained Systems via Constraint Propagation Analysis —

Hisaaki Shinkai Computataional Science Div., RIKEN Institute, Japan hshinkai@riken.jp

http://atlas.riken.go.jp/ ∼ shinkai/

Outline

• Introduction to Numerical Relativity

• Why mathematically equivalent eqs produce different numerical stability?

• Three approaches: (1) ADM/BSSN, (2) hyperbolic form. (3) attractor systems

• Proposals : A unified treatment as Adjusted Systems

Ref:

http://xxx.lanl.gov/abs/gr-qc/0209111

a review article, in print. (Nova Science Publ.)

at Minisymposium ”Numerical Methods for PDEs with Constraints”

in The 5th International Congress on Industrial and Applied Mathematics, Sydney, 2003 July.

Plan of the talk Control Constraints: H. Shinkai

1. Introduction: General Relativity and Numerical Relativity 2. Formulation problem of Numerical Relativity

(0) Arnowitt-Deser-Misner

(1) Baumgarte-Shapiro-Shibata-Nakamura formulation (2) Hyperbolic formulations

(3) Attractor systems 3. “Adjusted Systems”

Asymptotically constrained system by adjusting evolution eqs.

based on Constraint Propagation analysis

4. Summary and Future Issues

The Einstein equations (General Relativity): R

_µν

+

¹₂

g

_µν

R + Λg

_µν

= 8πGT

_µν

= Physics of strong gravity.

• gravitational waves from colliding black holes, neutron stars, supernovae, ...

• cosmology, higher-dimensional models, singularity, ...

• relativistic phenomena at active galactic nuclei, ...

What are the difficulties?

• for 10-component metric, highly nonlinear PDEs. (4 elliptic/6 hyperbolic in 3+1)

• free to choose cooordinates, gauge conditions, and even for decomposition of the space-time (3+1 or 2+2 or whatever).

• has singularity in its nature.

Solve it using computers, anyway !

Numerical Relativity = Solve the Einstein equations numerically.

over 20 years history, but still looking for a recipe ...

Toward Direct Detection of Gravitational Wave

GW is produced by coalescing Black-holes and/or Neutron Stars

Laser Interferometers

JAPAN 300m 2000- USA 4Km/2Km 2002- GermanyUK 600m 2002- ItalyFrance 3Km 2003-

~ mins (~ 1000 rot. )

?

~ 10 m sec

~ milli sec

INSPIRAL COALESCE BLACKHOLE FORMATION

Innermost Stable

Circular Orbit?

Post Newtonian Approx.

Numerical Relativity

BH. Perturbation

Numerical Relativity – open issues

0. How to foliate space-time

Cauchy (3 + 1), Hyperboloidal (3 + 1), characteristic (2 + 2), or combined?

⇒ if the foliation is (3 + 1), then · · · 1. How to prepare the initial data

Theoretical: Proper formulation for solving constraints? How to prepare realistic initial data?

Effects of background gravitational waves?

Connection to the post-Newtonian approximation?

Numerical: Techniques for solving coupled elliptic equations? Appropriate boundary conditions?

2. How to evolve the data

Theoretical: Free evolution or constrained evolution?

Proper formulation for the evolution equations? ⇒ THIS TALK!

Suitable slicing conditions (gauge conditions)?

Numerical: Techniques for solving the evolution equations? Appropriate boundary treatments?

Singularity excision techniques? Matter and shock surface treatments?

Parallelization of the code?

3. How to extract the physical information

Theoretical: Gravitational wave extraction? Connection to other approximations?

Numerical: Identification of black hole horizons? Visualization of simulations?

Numerical Relativity groups around the world

RIKEN, Japan

PennState

Kyoto, Japan

Tokyo, Japan

AEI, Germany

Southampton, UK Meudon, France

SISSA, Italy SouthAfrica

Pittsburgh Caltech

Vancouver Cornell

Louisiana Texas

UNAM, Mexico

Illinois

Monash, Australia

We are here

Numerical Relativity groups around the world

RIKEN, Japan

PennState

Kyoto, Japan

Tokyo, Japan

AEI, Germany

Southampton, UK Meudon, France

SISSA, Italy SouthAfrica

Pittsburgh Caltech

Vancouver Cornell

Louisiana Texas

UNAM, Mexico

Illinois

Monash, Australia

We are here

Galapagos Island

Plan of the talk Control Constraints: H. Shinkai

1. Introduction: General Relativity and Numerical Relativity 2. Formulation problem of Numerical Relativity

(0) Arnowitt-Deser-Misner

(1) Baumgarte-Shapiro-Shibata-Nakamura formulation (2) Hyperbolic formulations

(3) Attractor systems 3. “Adjusted Systems”

Asymptotically constrained system by adjusting evolution eqs.

based on Constraint Propagation analysis

4. Summary and Future Issues

strategy 0 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 linesurface normal line coordinate constant line

Nⁱ

lapse function, N

shift vector, N shift vector, Nⁱ

t = constant hypersurface t = constant hypersurface

Maxwell eqs. ADM Einstein eq.

constraints div E = 4πρ div B = 0

(3)R + (trK)² −K_ijK^ij = 2κρ_H + 2Λ D_jK^j_i − D_itrK = κJ_i

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 formulation of the Einstein eqs. for long-term stable & accurate simulation?

Many (too many) trials and errors, not yet a definit recipe.

time time

errorerror

Blow up Blow up

t=0

Constrained Surface

(satisfies Einstein's constraints)

Best formulation of the Einstein eqs. for long-term stable & accurate simulation?

Many (too many) trials and errors, not yet a definit recipe.

time time

errorerror

Blow up

Blow up Blow upBlow up

ADM ADM

BSSN BSSN

Mathematically equivalent formulations, 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

By adding constraints in RHS, we can kill error-growing modes

⇒ How can we understand the features systematically?

80s 90s 2000s

A D M

Shibata-Nakamura

95

Baumgarte-Shapiro

99

Nakamura-Oohara

87

Bona-Masso

92

Anderson-York

99

ChoquetBruhat-York

95-97

Frittelli-Reula

96 62

Ashtekar

86

Yoneda-Shinkai

99

Kidder-Scheel -Teukolsky

01

lambda-system

99

Alcubierre

97

Iriondo-Leguizamon-Reula 97

80s 90s 2000s

A D M

Shibata-Nakamura

95

Baumgarte-Shapiro

99

Nakamura-Oohara

87

Bona-Masso

92

Anderson-York

99

ChoquetBruhat-York

95-97

Frittelli-Reula

96 62

Ashtekar

86

Yoneda-Shinkai

99

Kidder-Scheel -Teukolsky

01

NCSA AEI

G-code

H-code BSSN-code

Cornell-Illinois

UWash

Hern

Caltech PennState

lambda-system

99

Shinkai-Yoneda Alcubierre

97

Nakamura-Oohara Shibata

Iriondo-Leguizamon-Reula 97

LSU

strategy 1 Shibata-Nakamura’s (Baumgarte-Shapiro’s) modifications to the standard ADM – define new variables (φ,γ˜_ij,K,A˜_ij,Γ˜ⁱ), instead of the ADM’s (γ_ij,K_ij) where

˜

γ_ij ≡ e^−4φγ_ij, A˜_ij ≡ e^−4φ(K_ij −(1/3)γ_ijK), Γ˜ⁱ ≡ Γ˜ⁱ_jkγ˜^jk, use momentum constraint in Γⁱ-eq., and impose det˜γ_ij = 1 during the evolutions.

– The set of evolution equations become (∂_t − L^β)φ = −(1/6)αK,

(∂_t − L^β)˜γ_ij = −2αA˜_ij,

(∂_t − L^β)K = αA˜_ijA˜^ij + (1/3)αK² − γ^ij(∇ⁱ∇^jα),

(∂_t − L^β) ˜A_ij = −e^−4φ(∇ⁱ∇^jα)^{T F} +e^−4φαR⁽³⁾_ij − e^−4φα(1/3)γ_ijR⁽³⁾ +α(KA˜_ij − 2 ˜A_ikA˜^k_j)

∂_tΓ˜ⁱ = −2(∂_jα) ˜A^ij − (4/3)α(∂_jK)˜γ^ij + 12αA˜^ji(∂_jφ) − 2αA˜_k^j(∂_jγ˜^ik) −2αΓ˜^k_ljA˜^j_kγ˜^il

−∂_j β^k∂_kγ˜^ij − γ˜^kj(∂_kβⁱ) − γ˜^ki(∂_kβ^j) + (2/3)˜γ^ij(∂_kβ^k)

R_ij = ∂_kΓ^k_ij − ∂_iΓ^k_kj + Γ^m_ijΓ^k_mk − Γ^m_kjΓ^k_mi =: ˜R_ij +R^φ_ij

R^φ_ij = −2 ˜D_iD˜_jφ− 2˜g_ijD˜^lD˜_lφ+ 4( ˜D_iφ)( ˜D_jφ)− 4˜g_ij( ˜D^lφ)( ˜D_lφ)

R˜_ij = −(1/2)˜g^lm∂_lmg˜_ij + ˜g_k(i∂_j)Γ˜^k + ˜Γ^kΓ˜_(ij)k + 2˜g^lmΓ˜^k_l(iΓ˜_j)km + ˜glmΓ˜^k_imΓ˜_klj – No explicit explanations why this formulation works better.

AEI group (2000): the replacement by momentum constraint is essential.

strategy 2 Apply a formulation which reveals a hyperbolicity explicitly.

For a first order partial differential equations on a vector u,

∂_t









u₁ u₂ ...







 =







 A







 ∂_x









u₁ u₂ ...









characteristic part

+ B









u₁ u₂ ...









lower order part if the eigenvalues of A are

weakly hyperbolic all real.

strongly hyperbolic all real and ∃ a complete set of eigenvalues.

symmetric hyperbolic if A is real and symmetric (Hermitian).

Symmetric hyp.

Symmetric hyp.

Strongly hyp.

Strongly hyp.

Weakly hyp.

Weakly hyp.

Expectations

– Wellposed behaviour

symmetric hyperbolic system =⇒ WELL-POSED , ||u(t)|| ≤ e^κt||u(0)||

– Better boundary treatments ⇐= ∃ characteristic field.

– known numerical techniques in Newtonian hydrodynamics.

80s 90s 2000s

A D M

Shibata-Nakamura

95

Baumgarte-Shapiro

99

Nakamura-Oohara

87

Bona-Masso

92

Anderson-York

99

ChoquetBruhat-York

95-97

Frittelli-Reula

96 62

Ashtekar

86

Yoneda-Shinkai

99

Kidder-Scheel -Teukolsky

01

NCSA AEI

G-code

H-code BSSN-code

Cornell-Illinois

UWash

Hern

Caltech PennState

lambda-system

99

Shinkai-Yoneda Alcubierre

97

Nakamura-Oohara Shibata

Iriondo-Leguizamon-Reula 97

LSU

80s 90s 2000s

A D M

Shibata-Nakamura

95

Baumgarte-Shapiro

99

Nakamura-Oohara

87

Bona-Masso

92

Anderson-York

99

ChoquetBruhat-York

95-97

Frittelli-Reula

96 62

Ashtekar

86

Yoneda-Shinkai

99

Kidder-Scheel -Teukolsky

01

NCSA AEI

G-code

H-code BSSN-code

Cornell-Illinois

UWash

Hern

Caltech PennState

lambda-system

99

adjusted-system

01

Shinkai-Yoneda Alcubierre

97

Nakamura-Oohara Shibata

Iriondo-Leguizamon-Reula 97

LSU Illinois

strategy 3 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)

time time errorerror

Blow up Blow up

Stabilize?

Stabilize?

?

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 sym- metric hyperbolic system.

method 2: Adjusted system (HS-Yoneda, 2000, 2001) – We can control the violation of constraints by ad-

justing constraints to EoM.

– Eigenvalue analysis of constraint propagation equations may prodict the violation of error.

– This idea is applicable even if the system is not symmetric hyperbolic. ⇒

for the ADM/BSSN formulation, too!!

Idea of λ-system

Brodbeck, Frittelli, H¨ubner and Reula, JMP40(99)909 We expect a system that is robust for controlling the violation of constraints

Recipe

1.

Prepare a symmetric hyperbolic evolution system

^∂tu = J∂_iu +K 2.

Introduce λ as an indicator of violation of constraint

which obeys dissipative eqs. of motion

∂_tλ = αC − βλ (α = 0, β > 0) 3.

Take a set of (u, λ) as dynamical variables

∂_t



u λ







A 0 F 0



∂_i



u λ





4.

Modify evolution eqs so as to form

a symmetric hyperbolic system

^∂t



u λ



 =



A F¯ F 0



∂_i



u λ





Remarks

• BFHR used a sym. hyp. formulation by Frittelli-Reula [PRL76(96)4667]

• The version for the Ashtekar formulation by HS-Yoneda [PRD60(99)101502]

for controlling the constraints or reality conditions or both.

• Succeeded in evolution of GW in planar spacetime using Ashtekar vars. [CQG18(2001)441]

• Do the recovered solutions represent true evolution? by Siebel-H¨ubner [PRD64(2001)024021]

Idea of “Adjusted system” and Our Conjecture

CQG18 (2001) 441, PRD 63 (2001) 120419, CQG 19 (2002) 1027 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(Ca, ∂}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.

The Adjusted system (essentials):

Purpose: Control the violation of constraints by reformulating the system so as to have a constrained surface an attractor.

Procedure: Add constraints to evolution eqs, and adjust its multipliers.

Theoretical support: Eigenvalue analysis of the constraint propagation equations.

Advantages: Available even if the base system is not a symmetric hyperbolic.

Advantages: Keep the number of the variable same with the original system.

Conjecture on Constraint Amplification Factors (CAFs):

∂_t









Cˆ₁ ...

Cˆ_N







 =









Constraint Propagation

Matrix

















Cˆ₁ ...

Cˆ_N







,

Eigenvalues = CAFs

We see more stable evolution, if CAFs have

(A) negative real-part (the constraints are forced to be diminished), or

(B) non-zero imaginary-part (the constraints are prop- agating away).

Example: the Maxwell equations

Yoneda HS, CQG 18 (2001) 441 Maxwell evolution equations.

∂_tE_i = c_i^jk∂_jB_k + P_iC_E +Q_iC_B,

∂_tB_i = −c_i^jk∂_jE_k +R_iC_E +S_iC_B,

C_E = ∂_iEⁱ ≈ 0, C_B = ∂_iBⁱ ≈ 0,















sym. hyp ⇔ P_i = Q_i = R_i = S_i = 0, strongly hyp ⇔ (P_i − S_i)² + 4R_iQ_i > 0, weakly hyp ⇔ (P_i −S_i)² + 4R_iQ_i ≥ 0 Constraint propagation equations

∂_tC_E = (∂_iPⁱ)C_E +Pⁱ(∂_iC_E) + (∂_iQⁱ)C_B +Qⁱ(∂_iC_B),

∂_tC_B = (∂_iRⁱ)C_E +Rⁱ(∂_iC_E) + (∂_iSⁱ)C_B + Sⁱ(∂_iC_B),















sym. hyp ⇔ Q_i = R_i,

strongly hyp ⇔ (P_i −S_i)² + 4R_iQ_i > 0, weakly hyp ⇔ (P_i −S_i)² + 4R_iQ_i ≥ 0 CAFs?

∂_t



Cˆ_E Cˆ_B



 =



∂_iPⁱ +Pⁱk_i ∂_iQⁱ + Qⁱk_i

∂_iRⁱ +Rⁱk_i ∂_iSⁱ + Sⁱk_i



∂_l



Cˆ_E Cˆ_B



 ≈



Pⁱk_i Qⁱk_i Rⁱk_i Sⁱk_i







Cˆ_E Cˆ_B



 =: T



Cˆ_E Cˆ_B





⇒ CAFs = (Pⁱk_i +Sⁱk_i ± (Pⁱk_i +Sⁱk_i)² + 4(Qⁱk_iR^jk_j −Pⁱk_iS^jk_j))/2 Therefore CAFs become negative-real when

Pⁱk_i +Sⁱk_i < 0, and Qⁱk_iR^jk_j − Pⁱk_iS^jk_j < 0

Plan of the talk Control Constraints: H. Shinkai

1. Introduction: General Relativity and Numerical Relativity 2. Formulation problem of Numerical Relativity

(0) Arnowitt-Deser-Misner

(1) Baumgarte-Shapiro-Shibata-Nakamura formulation (2) Hyperbolic formulations

(3) Attractor systems 3. “Adjusted Systems”

Asymptotically constrained system by adjusting evolution eqs.

based on Constraint Propagation analysis

4. Summary and Future Issues

3

Adjusted ADM systems

We adjust the standard ADM system using constraints as:

∂_tγ_ij = −2αK_ij +∇ⁱβ_j + ∇^jβ_i, (1) +P_ijH +Q^k_ijM^k +p^k_ij(∇^kH) + q^kl_ij(∇^kM^l), (2)

∂_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,(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)

Original ADM vs Standard ADM

Original ADM (ADM, 1962) the pair of (h_ij, π^ij).

L = √

−gR = √

hN[⁽³⁾R − K² + K_ijK^ij], π^ij = ∂L

∂h˙_ij = √

h(K^ij − Kh^ij), H = π^ijh˙_ij − L



























∂_th_ij = δH

δπ^ij = 2 N

√h(π_ij − 1

2h_ijπ) + 2D_(iN_j),

∂_tπ^ij = −δH

δh_ij = −√

hN(⁽³⁾R^ij − 1 2

(3)Rh^ij) + 1 2

√N

hh^ij(π_mnπ^mn − 1

2π²) − 2 N

√h(πⁱⁿπ_n^j − 1

2ππ^ij) +√

h(DⁱD^jN − h^ijD^mD_mN) + √

hD_m(h^−1/2N^mπ^ij) − 2π^m(iD_mN^j)

Standard ADM (York, 1979) the pair of (h_ij, K_ij).







∂_th_ij = −2N K_ij + D_jN_i + D_iN_j,

∂_tK_ij = N( ⁽³⁾R_ij + KK_ij) −2N K_ilK^l_j − D_iD_jN + (D_jN^m)K_mi + (D_iN^m)K_mj +N^mD_mK_ij

In this converting process, H was used.

That is, the standard ADM is already adjusted.

Constraint propagation of ADM systems

(1) Original ADM vs Standard ADM

With the adjustment R_ij = κ₁αγ_ij and other multiplier zero, whereκ₁ =







0 the standard ADM

−1/4 the original ADM

• The constraint propagation eqs keep the first-order form (cf Frittelli, PRD55(97)5992):

∂_t



 H Mⁱ







 β^l −2αγ^jl

−(1/2)αδ_i^l +R^l_i −δ_i^lR β^lδ_i^j



∂_l



 H M^j



. (1)

The eigenvalues of the characteristic matrix:

λ^l = (β^l, β^l, β^l ± α²γ^ll(1 + 4κ₁)) The hyperbolicity of (1):















symmetric hyperbolic when κ₁ = 3/2

strongly hyperbolic when α²γ^ll(1 + 4κ₁) > 0 weakly hyperbolic when α²γ^ll(1 + 4κ₁) ≥ 0

• On the Minkowskii background metric, the linear order terms of the Fourier-transformed constraint propagation equations gives the eigenvalues

Λ^l = (0,0,± −k²(1 + 4κ₁)).

That is,







(two 0s, two pure imaginary) for the standard ADM BETTER STABILITY

(four 0s) for the original ADM

Example 1: standard ADM vs original ADM (in Schwarzschild coordinate)

- 1 -0.5 0 0.5 1

0 5 1 0 1 5 2 0

no adjustments (standard ADM)

Real / Imaginary parts of Eigenvalues (AF)

rsch

( a )

-0.5 0 0.5

0 5 1 0 1 5 2 0

original ADM (κ

F= - 1/4)

rsch

( b )

Real / Imaginary parts of Eigenvalues (AF)

Figure 1: Amplification factors (AFs, eigenvalues of homogenized constraint propagation equations) are shown for the standard Schwarzschild coordinate, with (a) no adjustments, i.e., standard ADM, (b) original ADM (κ_F = −1/4). The solid lines and the dotted lines with circles are real parts and imaginary parts, respectively. They are four lines each, but actually the two eigenvalues are zero for all cases. Plotting range is 2< r ≤20 using Schwarzschild radial coordinate. We set k = 1, l = 2, and m= 2 throughout the article.

∂_tγ_ij = −2αK_ij +∇ⁱβ_j + ∇^jβ_i,

∂_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 + κ_Fαγ_ijH,

Constraint propagation of ADM systems

(2) Detweiler’s system

Detweiler’s modification to ADM [PRD35(87)1095] can be realized in our notation as:

P_ij = −Lα³γ_ij,

R_ij = Lα³(K_ij − (1/3)Kγ_ij),

S_ij^k = Lα²[3(∂_(iα)δ_j^k₎ − (∂_lα)γ_ijγ^kl],

s^kl_ij = Lα³[2δ_(i^kδ_j)^l − (1/3)γ_ijγ^kl], and else zero, where L is a constant.

• This adjustment does not make constraint propagation equation in the first order form, so that we can not discuss the hyperbolicity nor the characteristic speed of the constraints.

• For the Minkowskii background spacetime, the adjusted constraint propagation equations with above choice of multiplier become

∂_tH = −2(∂_jMj) + 4L(∂_j∂_jH),

∂_tMⁱ = −(1/2)(∂_iH) + (L/2)(∂_k∂_kMⁱ) + (L/6)(∂_i∂_kM^k).

Constraint Amp. Factors (the eigenvalues of their Fourier expression) are

Λ^l = (−(L/2)k²(multiplicity 2),−(7L/3)k² ± (1/3) k²(−9 + 25L²k²).) This indicates negative real eigenvalues if we chose small positive L.

Detweiler’s criteria vs Our criteria

• Detweiler calculated the L2 norm of the constraints, C_α, over the 3-hypersurface and imposed its negative definiteness of its evolution,

Detweiler’s criteria ⇔ ∂_t

α C_α² dV < 0,

This is rewritten by supposing the constraint propagation to be ∂_tCˆ_α = A_α^βCˆ_β in the Fourier components,

⇔ ∂_t

α

Cˆ_αC¯ˆ_α dV =

α A_α^βCˆ_βC¯ˆ_α + ˆC_αA¯_α^βC¯ˆ_β dV < 0, ∀ non zero Cˆ_α

⇔ eigenvalues of (A +A^†) are all negative for ∀k.

• Our criteria is that the eigenvalues of A are all negative. Therefore,

Our criteria Detweiler’s criteria

• We remark that Detweiler’s truncations on higher order terms in C-norm corresponds our perturbative analysis, both based on the idea that the deviations from constraint surface (the errors expressed non-zero constraint value) are initially small.

Example 2: Detweiler-type adjusted (in Schwarzschild coord.)

- 1 -0.5 0 0.5 1

0 5 1 0 1 5 2 0

Detweiler type, κ

L = + 1/2 ( b )

Real / Imaginary parts of Eigenvalues (AF)

rsch

- 1 -0.5 0 0.5 1

0 5 1 0 1 5 2 0

Detweiler type, κ

L = - 1/2

Real / Imaginary parts of Eigenvalues (AF)

( c )

rsch

Figure 2: Amplification factors of the standard Schwarzschild coordinate, with Detweiler type adjustments. Multipliers used in the plot are (b) κ_L= +1/2, and (c) κ_L =−1/2.

∂_tγ_ij = (original terms) +P_ijH,

∂_tK_ij = (original terms) +R_ijH + S^k_ijM^k + s^kl_ij(∇^kM^l),

where P_ij = −κ_Lα³γ_ij, R_ij = κ_Lα³(K_ij − (1/3)Kγ_ij),

S^k_ij = κ_Lα²[3(∂_(iα)δ_j)^k − (∂_lα)γ_ijγ^kl], s^kl_ij = κ_Lα³[δ_(i^kδ_j)^l − (1/3)γ_ijγ^kl],

Example 3: standard ADM (in isotropic/iEF coord.)

- 1 -0.5 0 0.5 1

0 5 1 0 1 5 2 0

isotropic coordinate, no adjustments (standard ADM) ( a )

Real / Imaginary parts of Eigenvalues (AF)

riso

- 1 -0.5 0 0.5 1 1.5 2

0 5 1 0 1 5 2 0

iEF coordinate, no adjustments (standard ADM) ( b )

Real / Imaginary parts of Eigenvalues (AF)

rsch

Figure 3: Comparison of amplification factors between different coordinate expressions for the standard ADM formulation (i.e.

no adjustments). Fig. (a) is for the isotropic coordinate (1), and the plotting range is 1/2 ≤ r_iso. Fig. (b) is for the iEF coordinate (1) and we plot lines on the t= 0 slice for each expression. The solid four lines and the dotted four lines with circles are real parts and imaginary parts, respectively.

Example 4: Detweiler-type adjusted (in iEF/PG coord.)

- 1 -0.5 0 0.5 1 1.5 2

0 5 1 0 1 5 2 0

iEF coordinate, Detweiler type κ

L=+0.5 ( b )

Real / Imaginary parts of Eigenvalues (AF)

rsch

- 1 -0.5 0 0.5 1 1.5 2

0 5 1 0 1 5 2 0

PG coordinate, Detweiler type κ

L=+0.5 ( c )

Real / Imaginary parts of Eigenvalues (AF)

rsch

Figure 4: Similar comparison for Detweiler adjustments. κ_L = +1/2 for all plots.