Outer Boundary Conditions for the Generalized Harmonic Einstein - - PowerPoint PPT Presentation

▶
outer boundary conditions for the generalized harmonic
SMART_READER_LITE
LIVE PREVIEW

Outer Boundary Conditions for the Generalized Harmonic Einstein - - PowerPoint PPT Presentation

Apr 29, 2023 46 likes •831 views

Outer Boundary Conditions for the Generalized Harmonic Einstein Equations: Stability and Accuracy Oliver Rinne Work with the Caltech-Cornell Numerical Relativity Collaboration Theoretical Astrophysics and Relativity, California Institute of

slide-1
SLIDE 1

university-logo

Outer Boundary Conditions for the Generalized Harmonic Einstein Equations: Stability and Accuracy

Oliver Rinne Work with the Caltech-Cornell Numerical Relativity Collaboration

Theoretical Astrophysics and Relativity, California Institute of Technology

From Geometry to Numerics, IHP , Paris, November 21, 2006

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 1 / 33

slide-2
SLIDE 2

university-logo

Outline

1

Introduction

2

Construction of boundary conditions

3

Stability analysis

4

Accuracy comparisons

5

Summary

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 2 / 33

slide-3
SLIDE 3

university-logo

Outline

1

Introduction

2

Construction of boundary conditions

3

Stability analysis

4

Accuracy comparisons

5

Summary

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 3 / 33

slide-4
SLIDE 4

university-logo

The initial-boundary value problem

Consider Einstein’s equations on compact spatial domain Ω with smooth outer boundary ∂Ω

Σ(0) ∂Ω × [0, t] Ω Σ(t) ∂t ni

Boundary conditions should

1

yield a well-posed initial-boundary value problem

2

be compatible with the constraints (constraint-preserving)

3

minimize reflections, control incoming gravitational radiation

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 4 / 33

slide-5
SLIDE 5

university-logo

The initial-boundary value problem

Consider Einstein’s equations on compact spatial domain Ω with smooth outer boundary ∂Ω

Σ(0) ∂Ω × [0, t] Ω Σ(t) ∂t ni

Boundary conditions should

1

yield a well-posed initial-boundary value problem

2

be compatible with the constraints (constraint-preserving)

3

minimize reflections, control incoming gravitational radiation

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 4 / 33

slide-6
SLIDE 6

university-logo

Previous work

[Friedrich & Nagy 1999] formulation that satisfies all three requirements for the fully nonlinear vacuum Einstein equations (tetrad-based, evolves Weyl tensor) Necessary conditions for well-posedness can be verified using pseudo-differential techniques (Fourier-Laplace analysis) [Stewart 1998, Calabrese & Sarbach 2003, Sarbach & Tiglio 2005, Kreiss & Winicour 2006, R 2006] Alternate approach to proving well-posedness via semigroup theory [Reula & Sarbach 2005, Nagy & Sarbach 2006] Improved absorbing boundary conditions [Lau 2004-5, Novak & Bonazzola 2004, Buchman & Sarbach 2006] Some alternatives: spatial compactification, Cauchy-characteristic and Cauchy-perturbative matching, hyperboloidal slices, . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33

slide-7
SLIDE 7

university-logo

Previous work

[Friedrich & Nagy 1999] formulation that satisfies all three requirements for the fully nonlinear vacuum Einstein equations (tetrad-based, evolves Weyl tensor) Necessary conditions for well-posedness can be verified using pseudo-differential techniques (Fourier-Laplace analysis) [Stewart 1998, Calabrese & Sarbach 2003, Sarbach & Tiglio 2005, Kreiss & Winicour 2006, R 2006] Alternate approach to proving well-posedness via semigroup theory [Reula & Sarbach 2005, Nagy & Sarbach 2006] Improved absorbing boundary conditions [Lau 2004-5, Novak & Bonazzola 2004, Buchman & Sarbach 2006] Some alternatives: spatial compactification, Cauchy-characteristic and Cauchy-perturbative matching, hyperboloidal slices, . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33

slide-8
SLIDE 8

university-logo

Previous work

[Friedrich & Nagy 1999] formulation that satisfies all three requirements for the fully nonlinear vacuum Einstein equations (tetrad-based, evolves Weyl tensor) Necessary conditions for well-posedness can be verified using pseudo-differential techniques (Fourier-Laplace analysis) [Stewart 1998, Calabrese & Sarbach 2003, Sarbach & Tiglio 2005, Kreiss & Winicour 2006, R 2006] Alternate approach to proving well-posedness via semigroup theory [Reula & Sarbach 2005, Nagy & Sarbach 2006] Improved absorbing boundary conditions [Lau 2004-5, Novak & Bonazzola 2004, Buchman & Sarbach 2006] Some alternatives: spatial compactification, Cauchy-characteristic and Cauchy-perturbative matching, hyperboloidal slices, . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33

slide-9
SLIDE 9

university-logo

Previous work

[Friedrich & Nagy 1999] formulation that satisfies all three requirements for the fully nonlinear vacuum Einstein equations (tetrad-based, evolves Weyl tensor) Necessary conditions for well-posedness can be verified using pseudo-differential techniques (Fourier-Laplace analysis) [Stewart 1998, Calabrese & Sarbach 2003, Sarbach & Tiglio 2005, Kreiss & Winicour 2006, R 2006] Alternate approach to proving well-posedness via semigroup theory [Reula & Sarbach 2005, Nagy & Sarbach 2006] Improved absorbing boundary conditions [Lau 2004-5, Novak & Bonazzola 2004, Buchman & Sarbach 2006] Some alternatives: spatial compactification, Cauchy-characteristic and Cauchy-perturbative matching, hyperboloidal slices, . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33

slide-10
SLIDE 10

university-logo

Previous work

[Friedrich & Nagy 1999] formulation that satisfies all three requirements for the fully nonlinear vacuum Einstein equations (tetrad-based, evolves Weyl tensor) Necessary conditions for well-posedness can be verified using pseudo-differential techniques (Fourier-Laplace analysis) [Stewart 1998, Calabrese & Sarbach 2003, Sarbach & Tiglio 2005, Kreiss & Winicour 2006, R 2006] Alternate approach to proving well-posedness via semigroup theory [Reula & Sarbach 2005, Nagy & Sarbach 2006] Improved absorbing boundary conditions [Lau 2004-5, Novak & Bonazzola 2004, Buchman & Sarbach 2006] Some alternatives: spatial compactification, Cauchy-characteristic and Cauchy-perturbative matching, hyperboloidal slices, . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33

slide-11
SLIDE 11

university-logo

(Generalized) harmonic gauge

Harmonic coordinates xa = 0 Principal part of Einstein equations becomes wave operator on metric ψab, 0 = Rab ≃ −1

2ψab

Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints Ca ≡ Ha − xa = Ha + Γab

b = 0

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33

slide-12
SLIDE 12

university-logo

(Generalized) harmonic gauge

Generalized harmonic coordinates [Friedrich 1985] xa = Ha(x, ψ) Principal part of Einstein equations becomes wave operator on metric ψab, 0 = Rab ≃ −1

2ψab

Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints Ca ≡ Ha − xa = Ha + Γab

b = 0

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33

slide-13
SLIDE 13

university-logo

(Generalized) harmonic gauge

Generalized harmonic coordinates [Friedrich 1985] xa = Ha(x, ψ) Principal part of Einstein equations becomes wave operator on metric ψab, 0 = Rab ≃ −1

2ψab

Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints Ca ≡ Ha − xa = Ha + Γab

b = 0

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33

slide-14
SLIDE 14

university-logo

(Generalized) harmonic gauge

Generalized harmonic coordinates [Friedrich 1985] xa = Ha(x, ψ) Principal part of Einstein equations becomes wave operator on metric ψab, 0 = Rab ≃ −1

2ψab

Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints Ca ≡ Ha − xa = Ha + Γab

b = 0

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33

slide-15
SLIDE 15

university-logo

(Generalized) harmonic gauge

Generalized harmonic coordinates [Friedrich 1985] xa = Ha(x, ψ) Principal part of Einstein equations becomes wave operator on metric ψab, 0 = Rab ≃ −1

2ψab

Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints Ca ≡ Ha − xa = Ha + Γab

b = 0

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33

slide-16
SLIDE 16

university-logo

First-order reduction

[Lindblom et al. 2006] Introduce new variables for first time and spatial derivatives of metric Πab ≡ −tc∂cψab, Φiab ≡ ∂iψab (ta normal to t = const. hypersurfaces, indices i, j, . . . = 1, 2, 3) New constraints Ciab ≡ ∂iψab − Φiab = 0, Cijab ≡ 2∂[iΦj]ab = 0 To principal parts, obtain ∂tψab ≃ 0, ∂tΠab ≃ Nk∂kΠab − Ngki∂kΦiab + γ2Nk∂kψab, ∂tΦiab ≃ Nk∂kΦiab − N∂iΠab + Nγ2∂iψab, (gab = ψab + tatb spatial metric, (∂t)a = Nta + Na lapse & shift)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 7 / 33

slide-17
SLIDE 17

university-logo

First-order reduction

[Lindblom et al. 2006] Introduce new variables for first time and spatial derivatives of metric Πab ≡ −tc∂cψab, Φiab ≡ ∂iψab (ta normal to t = const. hypersurfaces, indices i, j, . . . = 1, 2, 3) New constraints Ciab ≡ ∂iψab − Φiab = 0, Cijab ≡ 2∂[iΦj]ab = 0 To principal parts, obtain ∂tψab ≃ 0, ∂tΠab ≃ Nk∂kΠab − Ngki∂kΦiab + γ2Nk∂kψab, ∂tΦiab ≃ Nk∂kΦiab − N∂iΠab + Nγ2∂iψab, (gab = ψab + tatb spatial metric, (∂t)a = Nta + Na lapse & shift)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 7 / 33

slide-18
SLIDE 18

university-logo

First-order reduction

[Lindblom et al. 2006] Introduce new variables for first time and spatial derivatives of metric Πab ≡ −tc∂cψab, Φiab ≡ ∂iψab (ta normal to t = const. hypersurfaces, indices i, j, . . . = 1, 2, 3) New constraints Ciab ≡ ∂iψab − Φiab = 0, Cijab ≡ 2∂[iΦj]ab = 0 To principal parts, obtain ∂tψab ≃ 0, ∂tΠab ≃ Nk∂kΠab − Ngki∂kΦiab + γ2Nk∂kψab, ∂tΦiab ≃ Nk∂kΦiab − N∂iΠab + Nγ2∂iψab, (gab = ψab + tatb spatial metric, (∂t)a = Nta + Na lapse & shift)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 7 / 33

slide-19
SLIDE 19

university-logo

Characteristic structure

System is symmetric hyperbolic, characteristic variables in direction ni are u0

ab = ψab,

speed 0, u1±

ab = Πab ± Φnab − γ2ψab,

speed − Nn ± N, u2

Aab = ΦAab,

speed − Nn (vn ≡ nivi, vA ≡ PAivi, boundary metric Pij ≡ gij − ninj) Note dependence of speeds on normal component Nn of shift

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 8 / 33

slide-20
SLIDE 20

university-logo

Characteristic structure

System is symmetric hyperbolic, characteristic variables in direction ni are u0

ab = ψab,

speed 0, u1±

ab = Πab ± Φnab − γ2ψab,

speed − Nn ± N, u2

Aab = ΦAab,

speed − Nn (vn ≡ nivi, vA ≡ PAivi, boundary metric Pij ≡ gij − ninj) Note dependence of speeds on normal component Nn of shift

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 8 / 33

slide-21
SLIDE 21

university-logo

Outline

1

Introduction

2

Construction of boundary conditions

3

Stability analysis

4

Accuracy comparisons

5

Summary

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 9 / 33

slide-22
SLIDE 22

university-logo

Constraint-preserving boundary conditions

Constraints obey subsidiary system Ca ≃ 0 Set incoming modes of this system to zero at the boundary (in contrast, [Kreiss & Winicour 2006] use Ca . = 0) Obtain conditions on normal derivatives of 4 components of main incoming fields u1−, PC cd

ab

∂nu1−

cd

. = (tangential derivatives), where PC is projection operator with rank 4 If Nn ˙ >0 then u2

Aab also need boundary conditions, obtained by

requiring CnAab . = 0 ⇒ ∂nΦAbc . = ∂AΦnbc

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 10 / 33

slide-23
SLIDE 23

university-logo

Constraint-preserving boundary conditions

Constraints obey subsidiary system Ca ≃ 0 Set incoming modes of this system to zero at the boundary (in contrast, [Kreiss & Winicour 2006] use Ca . = 0) Obtain conditions on normal derivatives of 4 components of main incoming fields u1−, PC cd

ab

∂nu1−

cd

. = (tangential derivatives), where PC is projection operator with rank 4 If Nn ˙ >0 then u2

Aab also need boundary conditions, obtained by

requiring CnAab . = 0 ⇒ ∂nΦAbc . = ∂AΦnbc

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 10 / 33

slide-24
SLIDE 24

university-logo

Constraint-preserving boundary conditions

Constraints obey subsidiary system Ca ≃ 0 Set incoming modes of this system to zero at the boundary (in contrast, [Kreiss & Winicour 2006] use Ca . = 0) Obtain conditions on normal derivatives of 4 components of main incoming fields u1−, PC cd

ab

∂nu1−

cd

. = (tangential derivatives), where PC is projection operator with rank 4 If Nn ˙ >0 then u2

Aab also need boundary conditions, obtained by

requiring CnAab . = 0 ⇒ ∂nΦAbc . = ∂AΦnbc

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 10 / 33

slide-25
SLIDE 25

university-logo

Constraint-preserving boundary conditions

Constraints obey subsidiary system Ca ≃ 0 Set incoming modes of this system to zero at the boundary (in contrast, [Kreiss & Winicour 2006] use Ca . = 0) Obtain conditions on normal derivatives of 4 components of main incoming fields u1−, PC cd

ab

∂nu1−

cd

. = (tangential derivatives), where PC is projection operator with rank 4 If Nn ˙ >0 then u2

Aab also need boundary conditions, obtained by

requiring CnAab . = 0 ⇒ ∂nΦAbc . = ∂AΦnbc

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 10 / 33

slide-26
SLIDE 26

university-logo

Physical boundary conditions

Incoming gravitational radiation ⇔ Newman-Penrose scalar Ψ0 = Cabcdlamblcmd, {la = (ta + na)/ √ 2, ka = (ta − na)/ √ 2, ma, ¯ ma} complex null tetrad We impose the BC Ψ0 . = 0 Rewrite as PP cd

ab ∂nu1− cd

. = (tangential derivatives) + hP

ab,

where PP has rank 2 and is orthogonal to PC Lowest level in a hierarchy of perfectly absorbing BCs for linearized gravitational waves [Luisa Buchman’s talk]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 11 / 33

slide-27
SLIDE 27

university-logo

Physical boundary conditions

Incoming gravitational radiation ⇔ Newman-Penrose scalar Ψ0 = Cabcdlamblcmd, {la = (ta + na)/ √ 2, ka = (ta − na)/ √ 2, ma, ¯ ma} complex null tetrad We impose the BC Ψ0 . = 0 Rewrite as PP cd

ab ∂nu1− cd

. = (tangential derivatives) + hP

ab,

where PP has rank 2 and is orthogonal to PC Lowest level in a hierarchy of perfectly absorbing BCs for linearized gravitational waves [Luisa Buchman’s talk]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 11 / 33

slide-28
SLIDE 28

university-logo

Physical boundary conditions

Incoming gravitational radiation ⇔ Newman-Penrose scalar Ψ0 = Cabcdlamblcmd, {la = (ta + na)/ √ 2, ka = (ta − na)/ √ 2, ma, ¯ ma} complex null tetrad We impose the BC Ψ0 . = hP Rewrite as PP cd

ab ∂nu1− cd

. = (tangential derivatives) + hP

ab,

where PP has rank 2 and is orthogonal to PC Lowest level in a hierarchy of perfectly absorbing BCs for linearized gravitational waves [Luisa Buchman’s talk]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 11 / 33

slide-29
SLIDE 29

university-logo

Physical boundary conditions

Incoming gravitational radiation ⇔ Newman-Penrose scalar Ψ0 = Cabcdlamblcmd, {la = (ta + na)/ √ 2, ka = (ta − na)/ √ 2, ma, ¯ ma} complex null tetrad We impose the BC Ψ0 . = hP Rewrite as PP cd

ab ∂nu1− cd

. = (tangential derivatives) + hP

ab,

where PP has rank 2 and is orthogonal to PC Lowest level in a hierarchy of perfectly absorbing BCs for linearized gravitational waves [Luisa Buchman’s talk]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 11 / 33

slide-30
SLIDE 30

university-logo

Physical boundary conditions

Incoming gravitational radiation ⇔ Newman-Penrose scalar Ψ0 = Cabcdlamblcmd, {la = (ta + na)/ √ 2, ka = (ta − na)/ √ 2, ma, ¯ ma} complex null tetrad We impose the BC Ψ0 . = hP Rewrite as PP cd

ab ∂nu1− cd

. = (tangential derivatives) + hP

ab,

where PP has rank 2 and is orthogonal to PC Lowest level in a hierarchy of perfectly absorbing BCs for linearized gravitational waves [Luisa Buchman’s talk]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 11 / 33

slide-31
SLIDE 31

university-logo

Gauge boundary conditions

Remaining gauge freedom xa → xa + ξa provided that ξa = 0 Induced metric change ψab → ψab − 2∂(aξb) Ideally, impose absorbing BC on ξa To leading order in inverse radius, a suitable BC is PG cd

ab

(u1−

cd + γ2ψab) .

= 0 where PG has rank 4 and PC + PP + PG = I

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 12 / 33

slide-32
SLIDE 32

university-logo

Gauge boundary conditions

Remaining gauge freedom xa → xa + ξa provided that ξa = 0 Induced metric change ψab → ψab − 2∂(aξb) Ideally, impose absorbing BC on ξa To leading order in inverse radius, a suitable BC is PG cd

ab

(u1−

cd + γ2ψab) .

= 0 where PG has rank 4 and PC + PP + PG = I

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 12 / 33

slide-33
SLIDE 33

university-logo

Gauge boundary conditions

Remaining gauge freedom xa → xa + ξa provided that ξa = 0 Induced metric change ψab → ψab − 2∂(aξb) Ideally, impose absorbing BC on ξa To leading order in inverse radius, a suitable BC is PG cd

ab

(u1−

cd + γ2ψab) .

= 0 where PG has rank 4 and PC + PP + PG = I

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 12 / 33

slide-34
SLIDE 34

university-logo

Gauge boundary conditions

Remaining gauge freedom xa → xa + ξa provided that ξa = 0 Induced metric change ψab → ψab − 2∂(aξb) Ideally, impose absorbing BC on ξa To leading order in inverse radius, a suitable BC is PG cd

ab

(u1−

cd + γ2ψab) .

= 0 where PG has rank 4 and PC + PP + PG = I

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 12 / 33

slide-35
SLIDE 35

university-logo

Gauge boundary conditions

Remaining gauge freedom xa → xa + ξa provided that ξa = 0 Induced metric change ψab → ψab − 2∂(aξb) Ideally, impose absorbing BC on ξa To leading order in inverse radius, a suitable BC is PG cd

ab

(u1−

cd + γ2ψab) .

= hG

cd

where PG has rank 4 and PC + PP + PG = I

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 12 / 33

slide-36
SLIDE 36

university-logo

Outline

1

Introduction

2

Construction of boundary conditions

3

Stability analysis

4

Accuracy comparisons

5

Summary

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 13 / 33

slide-37
SLIDE 37

university-logo

Fourier-Laplace analysis

Consider high-frequency perturbations about any given spacetime Obtain linear symmetric hyperbolic system with constant coefficients Solve by Laplace transform in time and Fourier transform in space Boundary conditions imply linear system of equations for integration constants Study zeros of its (complex) determinant ⇒ necessary conditions for well-posedness (determinant condition and Kreiss condition) GH system satisfies both conditions [R 2006]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 14 / 33

slide-38
SLIDE 38

university-logo

Fourier-Laplace analysis

Consider high-frequency perturbations about any given spacetime Obtain linear symmetric hyperbolic system with constant coefficients Solve by Laplace transform in time and Fourier transform in space Boundary conditions imply linear system of equations for integration constants Study zeros of its (complex) determinant ⇒ necessary conditions for well-posedness (determinant condition and Kreiss condition) GH system satisfies both conditions [R 2006]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 14 / 33

slide-39
SLIDE 39

university-logo

Fourier-Laplace analysis

Consider high-frequency perturbations about any given spacetime Obtain linear symmetric hyperbolic system with constant coefficients Solve by Laplace transform in time and Fourier transform in space Boundary conditions imply linear system of equations for integration constants Study zeros of its (complex) determinant ⇒ necessary conditions for well-posedness (determinant condition and Kreiss condition) GH system satisfies both conditions [R 2006]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 14 / 33

slide-40
SLIDE 40

university-logo

Fourier-Laplace analysis

Consider high-frequency perturbations about any given spacetime Obtain linear symmetric hyperbolic system with constant coefficients Solve by Laplace transform in time and Fourier transform in space Boundary conditions imply linear system of equations for integration constants Study zeros of its (complex) determinant ⇒ necessary conditions for well-posedness (determinant condition and Kreiss condition) GH system satisfies both conditions [R 2006]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 14 / 33

slide-41
SLIDE 41

university-logo

Fourier-Laplace analysis

Consider high-frequency perturbations about any given spacetime Obtain linear symmetric hyperbolic system with constant coefficients Solve by Laplace transform in time and Fourier transform in space Boundary conditions imply linear system of equations for integration constants Study zeros of its (complex) determinant ⇒ necessary conditions for well-posedness (determinant condition and Kreiss condition) GH system satisfies both conditions [R 2006]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 14 / 33

slide-42
SLIDE 42

university-logo

Fourier-Laplace analysis

Consider high-frequency perturbations about any given spacetime Obtain linear symmetric hyperbolic system with constant coefficients Solve by Laplace transform in time and Fourier transform in space Boundary conditions imply linear system of equations for integration constants Study zeros of its (complex) determinant ⇒ necessary conditions for well-posedness (determinant condition and Kreiss condition) GH system satisfies both conditions [R 2006]

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 14 / 33

slide-43
SLIDE 43

university-logo

Towards sufficient conditions

Kreiss condition implies that solution can be estimated in terms of boundary data (boundary-stable) One would also like to control

source terms (well-posedness in the generalized sense) initial data (well-posedness)

Proof via symmetrizer construction [Kreiss 1970] Technique not applicable to boundary conditions of differential type Can show that system is free of weak instabilities with polynomial time dependence

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 15 / 33

slide-44
SLIDE 44

university-logo

Towards sufficient conditions

Kreiss condition implies that solution can be estimated in terms of boundary data (boundary-stable) One would also like to control

source terms (well-posedness in the generalized sense) initial data (well-posedness)

Proof via symmetrizer construction [Kreiss 1970] Technique not applicable to boundary conditions of differential type Can show that system is free of weak instabilities with polynomial time dependence

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 15 / 33

slide-45
SLIDE 45

university-logo

Towards sufficient conditions

Kreiss condition implies that solution can be estimated in terms of boundary data (boundary-stable) One would also like to control

source terms (well-posedness in the generalized sense) initial data (well-posedness)

Proof via symmetrizer construction [Kreiss 1970] Technique not applicable to boundary conditions of differential type Can show that system is free of weak instabilities with polynomial time dependence

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 15 / 33

slide-46
SLIDE 46

university-logo

Towards sufficient conditions

Kreiss condition implies that solution can be estimated in terms of boundary data (boundary-stable) One would also like to control

source terms (well-posedness in the generalized sense) initial data (well-posedness)

Proof via symmetrizer construction [Kreiss 1970] Technique not applicable to boundary conditions of differential type Can show that system is free of weak instabilities with polynomial time dependence

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 15 / 33

slide-47
SLIDE 47

university-logo

Towards sufficient conditions

Kreiss condition implies that solution can be estimated in terms of boundary data (boundary-stable) One would also like to control

source terms (well-posedness in the generalized sense) initial data (well-posedness)

Proof via symmetrizer construction [Kreiss 1970] Technique not applicable to boundary conditions of differential type Can show that system is free of weak instabilities with polynomial time dependence

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 15 / 33

slide-48
SLIDE 48

university-logo

Numerical robust stability test

Consider fixed background solution (Minkowski or Schwarzschild) Add small random perturbations to initial data, boundary data and right-hand-sides of evolution equations Evolve on domain T 2 × R, impose BCs in transverse direction Pseudospectral collocation method [Caltech-Cornell Spectral Einstein Code] Monitor error (deviation from background solution) and constraint violations

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 16 / 33

slide-49
SLIDE 49

university-logo

Numerical robust stability test

Consider fixed background solution (Minkowski or Schwarzschild) Add small random perturbations to initial data, boundary data and right-hand-sides of evolution equations Evolve on domain T 2 × R, impose BCs in transverse direction Pseudospectral collocation method [Caltech-Cornell Spectral Einstein Code] Monitor error (deviation from background solution) and constraint violations

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 16 / 33

slide-50
SLIDE 50

university-logo

Numerical robust stability test

Consider fixed background solution (Minkowski or Schwarzschild) Add small random perturbations to initial data, boundary data and right-hand-sides of evolution equations Evolve on domain T 2 × R, impose BCs in transverse direction Pseudospectral collocation method [Caltech-Cornell Spectral Einstein Code] Monitor error (deviation from background solution) and constraint violations

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 16 / 33

slide-51
SLIDE 51

university-logo

Numerical robust stability test

Consider fixed background solution (Minkowski or Schwarzschild) Add small random perturbations to initial data, boundary data and right-hand-sides of evolution equations Evolve on domain T 2 × R, impose BCs in transverse direction Pseudospectral collocation method [Caltech-Cornell Spectral Einstein Code] Monitor error (deviation from background solution) and constraint violations

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 16 / 33

slide-52
SLIDE 52

university-logo

Numerical robust stability test

Consider fixed background solution (Minkowski or Schwarzschild) Add small random perturbations to initial data, boundary data and right-hand-sides of evolution equations Evolve on domain T 2 × R, impose BCs in transverse direction Pseudospectral collocation method [Caltech-Cornell Spectral Einstein Code] Monitor error (deviation from background solution) and constraint violations

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 16 / 33

slide-53
SLIDE 53

university-logo

Flat space without shift

200 400 600 800 1000

t

2×10

  • 10

4×10

  • 10

6×10

  • 10

8×10

  • 10

|| E ||2

9 points 15 points 21 points 27 points

N

i = (0, 0, 0) 200 400 600 800 1000

t

1×10

  • 9

1×10

  • 8

|| C ||2

9 points 15 points 21 points 27 points

N

i = (0, 0, 0)

Random data amplitude 10−10

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 17 / 33

slide-54
SLIDE 54

university-logo

Flat space with constant shift

200 400 600 800 1000

t

0.0 5.0×10

  • 10

1.0×10

  • 9

1.5×10

  • 9

2.0×10

  • 9

|| E ||2

9 points 15 points 21 points 27 points

N

i = (0.5, 0.5, 0) 200 400 600 800 1000

t

1×10

  • 9

1×10

  • 8

|| C ||2

9 points 15 points 21 points 27 points

N

i = (0.5, 0.5, 0)

Random data amplitude 10−10

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 18 / 33

slide-55
SLIDE 55

university-logo

Schwarzschild

200 400 600 800 1000

t / M

1×10

  • 5

2×10

  • 5

3×10

  • 5

4×10

  • 5

5×10

  • 5

|| E ||2

Nr = 15, L = 11 Nr = 21, L = 15 Nr = 27, L = 19 200 400 600 800 1000

t / M

1×10

  • 5

2×10

  • 5

3×10

  • 5

4×10

  • 5

|| C ||2

Nr = 15, L = 11 Nr = 21, L = 15 Nr = 27, L = 19

Random data amplitude 10−6

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 19 / 33

slide-56
SLIDE 56

university-logo

Outline

1

Introduction

2

Construction of boundary conditions

3

Stability analysis

4

Accuracy comparisons

5

Summary

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 20 / 33

slide-57
SLIDE 57

university-logo

Some alternative boundary treatments

Freezing all the incoming fields u1−

ab

. = 0 (and u2

Aab

. = 0 if Nn ˙ >0) Sommerfeld boundary conditions (popular for BSSN formulation), for spherical boundary of radius r = R, (∂t + ∂r + 1

R)ψab .

= 0 Spatial compactification [Pretorius 2005]

Choose mapping r → x(r) that maps spatial infinity to a finite coordinate location, e.g. x = arctan r Discretize uniformly in x Apply low-pass frequency filter to damp waves as they travel out

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 21 / 33

slide-58
SLIDE 58

university-logo

Some alternative boundary treatments

Freezing all the incoming fields u1−

ab

. = 0 (and u2

Aab

. = 0 if Nn ˙ >0) Sommerfeld boundary conditions (popular for BSSN formulation), for spherical boundary of radius r = R, (∂t + ∂r + 1

R)ψab .

= 0 Spatial compactification [Pretorius 2005]

Choose mapping r → x(r) that maps spatial infinity to a finite coordinate location, e.g. x = arctan r Discretize uniformly in x Apply low-pass frequency filter to damp waves as they travel out

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 21 / 33

slide-59
SLIDE 59

university-logo

Some alternative boundary treatments

Freezing all the incoming fields u1−

ab

. = 0 (and u2

Aab

. = 0 if Nn ˙ >0) Sommerfeld boundary conditions (popular for BSSN formulation), for spherical boundary of radius r = R, (∂t + ∂r + 1

R)ψab .

= 0 Spatial compactification [Pretorius 2005]

Choose mapping r → x(r) that maps spatial infinity to a finite coordinate location, e.g. x = arctan r Discretize uniformly in x Apply low-pass frequency filter to damp waves as they travel out

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 21 / 33

slide-60
SLIDE 60

university-logo

Our test problem

[Ongoing work with Lee Lindblom and Mark Scheel] Background solution: Schwarzschild black hole (mass M = 1) Add outgoing quadrupole wave perturbation [Teukolsky 1982], amplitude 4 × 10−3 (odd-parity) Evolve on a spherical shell extending from r = 1.9 (just inside the horizon) out to

R = 1000 (reference solution) R = 41.9, 81.9, . . .

On the smaller domain, either impose the boundary conditions described in this talk or apply one of the alternative methods Compute difference of the two numerical solutions, compare in- and outgoing radiation (Ψ0 and Ψ4), . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 22 / 33

slide-61
SLIDE 61

university-logo

Our test problem

[Ongoing work with Lee Lindblom and Mark Scheel] Background solution: Schwarzschild black hole (mass M = 1) Add outgoing quadrupole wave perturbation [Teukolsky 1982], amplitude 4 × 10−3 (odd-parity) Evolve on a spherical shell extending from r = 1.9 (just inside the horizon) out to

R = 1000 (reference solution) R = 41.9, 81.9, . . .

On the smaller domain, either impose the boundary conditions described in this talk or apply one of the alternative methods Compute difference of the two numerical solutions, compare in- and outgoing radiation (Ψ0 and Ψ4), . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 22 / 33

slide-62
SLIDE 62

university-logo

Our test problem

[Ongoing work with Lee Lindblom and Mark Scheel] Background solution: Schwarzschild black hole (mass M = 1) Add outgoing quadrupole wave perturbation [Teukolsky 1982], amplitude 4 × 10−3 (odd-parity) Evolve on a spherical shell extending from r = 1.9 (just inside the horizon) out to

R = 1000 (reference solution) R = 41.9, 81.9, . . .

On the smaller domain, either impose the boundary conditions described in this talk or apply one of the alternative methods Compute difference of the two numerical solutions, compare in- and outgoing radiation (Ψ0 and Ψ4), . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 22 / 33

slide-63
SLIDE 63

university-logo

Our test problem

[Ongoing work with Lee Lindblom and Mark Scheel] Background solution: Schwarzschild black hole (mass M = 1) Add outgoing quadrupole wave perturbation [Teukolsky 1982], amplitude 4 × 10−3 (odd-parity) Evolve on a spherical shell extending from r = 1.9 (just inside the horizon) out to

R = 1000 (reference solution) R = 41.9, 81.9, . . .

On the smaller domain, either impose the boundary conditions described in this talk or apply one of the alternative methods Compute difference of the two numerical solutions, compare in- and outgoing radiation (Ψ0 and Ψ4), . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 22 / 33

slide-64
SLIDE 64

university-logo

Our test problem

[Ongoing work with Lee Lindblom and Mark Scheel] Background solution: Schwarzschild black hole (mass M = 1) Add outgoing quadrupole wave perturbation [Teukolsky 1982], amplitude 4 × 10−3 (odd-parity) Evolve on a spherical shell extending from r = 1.9 (just inside the horizon) out to

R = 1000 (reference solution) R = 41.9, 81.9, . . .

On the smaller domain, either impose the boundary conditions described in this talk or apply one of the alternative methods Compute difference of the two numerical solutions, compare in- and outgoing radiation (Ψ0 and Ψ4), . . .

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 22 / 33

slide-65
SLIDE 65

university-logo

Old ∗ (solid) vs. new (dotted) CPBCs ( ∗ without the γ2ψ term in the gauge BCs)

200 400 600 800 1000

t / M

10

  • 8

10

  • 7

10

  • 6

10

  • 5

10

  • 4

10

  • 3

||△U ||∞

200 400 600 800 1000

t / M

10

  • 8

10

  • 7

10

  • 6

10

  • 5

10

  • 4

10

  • 3

|| C ||∞

R = 41.9, (Nr, L) = (21, 8), (31, 10), (41, 12), (51, 41)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 23 / 33

slide-66
SLIDE 66

university-logo

Freezing (solid) vs. new CP (dotted) BCs

200 400 600 800 1000

t / M

10

  • 8

10

  • 7

10

  • 6

10

  • 5

10

  • 4

10

  • 3

10

  • 2

||△U ||∞

200 400 600 800 1000

t / M

10

  • 8

10

  • 7

10

  • 6

10

  • 5

10

  • 4

10

  • 3

|| C ||∞

R = 41.9, (Nr, L) = (21, 8), (31, 10), (41, 12), (51, 41)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 24 / 33

slide-67
SLIDE 67

university-logo

Sommerfeld (solid) vs. new CP (dotted) BCs

200 400 600 800 1000

t / M

10

  • 8

10

  • 7

10

  • 6

10

  • 5

10

  • 4

10

  • 3

10

  • 2

||△U ||∞

200 400 600 800 1000

t / M

10

  • 9

10

  • 8

10

  • 7

10

  • 6

10

  • 5

10

  • 4

10

  • 3

|| C ||∞

R = 41.9, (Nr, L) = (21, 8), (31, 10), (41, 12), (51, 41)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 25 / 33

slide-68
SLIDE 68

university-logo

Tan compactification with various filters vs. new CPBCs

200 400 600 800 1000

t / M

10

  • 8

10

  • 6

10

  • 4

10

  • 2

||△U ||∞

200 400 600 800 1000

t / M

10

  • 8

10

  • 6

10

  • 4

10

  • 2

|| C ||∞

New constraint-preserving BCs Kreiss-Oliger filter (ǫ = 1) applied to RHS Hesthaven filter (σ = 0.76, p = 13) applied to RHS Kreiss-Oliger filter (ǫ = 0.25) applied to solution Hesthaven filter (σ = 0.76, p = 13) applied to solution

R = 41.9, (Nr, L) = (51, 14)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 26 / 33

slide-69
SLIDE 69

university-logo

Tan compactification with best filter (solid) vs. new CPBCs (dotted)

200 400 600 800 1000

t / M

10

  • 8

10

  • 6

10

  • 4

10

  • 2

||△U ||∞

200 400 600 800 1000

t / M

10

  • 8

10

  • 6

10

  • 4

10

  • 2

|| C ||∞

Hesthaven filter applied to solution, R = 41.9, (Nr, L) = (21, 8), (31, 10), (41, 12), (51, 41)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 27 / 33

slide-70
SLIDE 70

university-logo

Accuracy of extracted Ψ4

200 400 600 800 1000

t / M

10

  • 13

10

  • 11

10

  • 9

10

  • 7

10

  • 5

∆Ψ4 / max |Ψ4| Sommerfeld (solid) vs. new CP (dotted)

200 400 600 800 1000

t / M

10

  • 13

10

  • 11

10

  • 9

10

  • 7

10

  • 5

10

  • 3

10

  • 1

∆Ψ4 / max |Ψ4| Best compactified (solid) vs. new CP (dotted)

R = 41.9, (Nr, L) = (31, 10), (51, 41)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 28 / 33

slide-71
SLIDE 71

university-logo

The reflection coefficient: theory vs. “experiment”

[Buchman & Sarbach 2006] predict for our CPBCs Ψ0/Ψ4 = 4

9(kR)−4 + O[(kR)−5]

1 2 3 4 5 6 7

k

10

  • 18

10

  • 16

10

  • 14

10

  • 12

10

  • 10

10

  • 8

10

  • 6

ψ0 ψ4 ψ4 * 4/9 (k R)

  • 4

R = 41.9

1 2 3 4 5 6 7

k

10

  • 18

10

  • 16

10

  • 14

10

  • 12

10

  • 10

10

  • 8

10

  • 6

ψ0 ψ4 ψ4 * 4/9 (k R)

  • 4

R = 121.9

(Nr, L) = (51, 14)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 29 / 33

slide-72
SLIDE 72

university-logo

Outline

1

Introduction

2

Construction of boundary conditions

3

Stability analysis

4

Accuracy comparisons

5

Summary

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 30 / 33

slide-73
SLIDE 73

university-logo

Summary

Constructed a set of constraint-preserving and radiation-controlling boundary conditions for the generalized harmonic Einstein equations Verified necessary conditions for well-posedness using the Fourier-Laplace technique, supported by numerical robust stability tests Numerical results indicate that our BCs cause significantly less reflections than alternate methods such as spatial compactification or Sommerfeld BCs

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 31 / 33

slide-74
SLIDE 74

university-logo

Summary

Constructed a set of constraint-preserving and radiation-controlling boundary conditions for the generalized harmonic Einstein equations Verified necessary conditions for well-posedness using the Fourier-Laplace technique, supported by numerical robust stability tests Numerical results indicate that our BCs cause significantly less reflections than alternate methods such as spatial compactification or Sommerfeld BCs

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 31 / 33

slide-75
SLIDE 75

university-logo

Summary

Constructed a set of constraint-preserving and radiation-controlling boundary conditions for the generalized harmonic Einstein equations Verified necessary conditions for well-posedness using the Fourier-Laplace technique, supported by numerical robust stability tests Numerical results indicate that our BCs cause significantly less reflections than alternate methods such as spatial compactification or Sommerfeld BCs

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 31 / 33

slide-76
SLIDE 76

university-logo

Some references

[Lindblom et al. 2006]

  • L. Lindblom, M. A. Scheel, L. E. Kidder, R. Owen, and O. Rinne

A new generalized harmonic evolution system

  • Class. Quantum Grav. 23(16) S447–S462

[R 2006]

  • O. Rinne

Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations

  • Class. Quantum Grav. 23(22) 6275–6300

[Kreiss & Winicour 2006] H.-O. Kreiss and J. Winicour Problems which are well-posed in a generalized sense with applications to the Einstein equations

  • Class. Quantum Grav. 23(16) S405–S420

[Sarbach & Buchman 2006] O. C. A. Sarbach & L. T. Buchman Towards absorbing outer boundaries in general relativity

  • Class. Quantum Grav. 23(23) 6709–6744

[Pretorius 2005]

  • F. Pretorius

Numerical relativity using a generalized harmonic decomposition

  • Class. Quantum Grav. 22(2) 425–451

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 32 / 33

slide-77
SLIDE 77

university-logo

Spatial compactification: details

Compactification map e.g. r(x) = R tan(πx/4R) (Tan mapping) r(x) =

  • x,

0 x < R R2/(2R − x), R x < 2R (Inverse mapping)

0.5 1 1.5 2 x / R 1 2 3 4 5 r / R Inverse Tan

Filter function e.g. f(k) = 1 − ǫ sin4(πk/2kmax), where 0 ǫ 1 (Kreiss-Oliger filter) f(k) = exp[−(k/σkmax)p], typically σ = 0.76, p = 13 (Hesthaven filter)

0.5 1 k / kmax 0.5 1 f K.-O. (ε = .25) K.-O. (ε = 1)

  • Hesth. (σ = .76, p = 13)

Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 33 / 33