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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

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 university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 1 / 33
Outline Introduction 1 Construction of boundary conditions 2 Stability analysis 3 Accuracy comparisons 4 Summary 5 university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 2 / 33
Outline Introduction 1 Construction of boundary conditions 2 Stability analysis 3 Accuracy comparisons 4 Summary 5 university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 3 / 33
The initial-boundary value problem Consider Einstein’s equations on compact spatial domain Ω with smooth outer boundary ∂ Ω n i Ω Σ( t ) ∂ t ∂ Ω × [ 0 , t ] Σ( 0 ) Boundary conditions should yield a well-posed initial-boundary value problem 1 be compatible with the constraints ( constraint-preserving ) 2 minimize reflections, control incoming gravitational radiation 3 university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 4 / 33
The initial-boundary value problem Consider Einstein’s equations on compact spatial domain Ω with smooth outer boundary ∂ Ω n i Ω Σ( t ) ∂ t ∂ Ω × [ 0 , t ] Σ( 0 ) Boundary conditions should yield a well-posed initial-boundary value problem 1 be compatible with the constraints ( constraint-preserving ) 2 minimize reflections, control incoming gravitational radiation 3 university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 4 / 33
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, . . . university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33
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, . . . university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33
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, . . . university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33
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, . . . university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33
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, . . . university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 5 / 33
(Generalized) harmonic gauge Harmonic coordinates � x a = 0 Principal part of Einstein equations becomes wave operator on metric ψ ab , 0 = R ab ≃ − 1 2 � ψ ab Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints b = 0 C a ≡ H a − � x a = H a + Γ ab university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33
(Generalized) harmonic gauge Generalized harmonic coordinates [Friedrich 1985] � x a = H a ( x , ψ ) Principal part of Einstein equations becomes wave operator on metric ψ ab , 0 = R ab ≃ − 1 2 � ψ ab Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints b = 0 C a ≡ H a − � x a = H a + Γ ab university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33
(Generalized) harmonic gauge Generalized harmonic coordinates [Friedrich 1985] � x a = H a ( x , ψ ) Principal part of Einstein equations becomes wave operator on metric ψ ab , 0 = R ab ≃ − 1 2 � ψ ab Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints b = 0 C a ≡ H a − � x a = H a + Γ ab university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33
(Generalized) harmonic gauge Generalized harmonic coordinates [Friedrich 1985] � x a = H a ( x , ψ ) Principal part of Einstein equations becomes wave operator on metric ψ ab , 0 = R ab ≃ − 1 2 � ψ ab Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints b = 0 C a ≡ H a − � x a = H a + Γ ab university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33
(Generalized) harmonic gauge Generalized harmonic coordinates [Friedrich 1985] � x a = H a ( x , ψ ) Principal part of Einstein equations becomes wave operator on metric ψ ab , 0 = R ab ≃ − 1 2 � ψ ab Symmetric hyperbolic system, Cauchy problem is well-posed [Choquet-Bruhat 1952] Subject to constraints b = 0 C a ≡ H a − � x a = H a + Γ ab university-logo Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 6 / 33
First-order reduction [Lindblom et al. 2006] Introduce new variables for first time and spatial derivatives of metric Π ab ≡ − t c ∂ c ψ ab , Φ iab ≡ ∂ i ψ ab ( t a normal to t = const . hypersurfaces, indices i , j , . . . = 1 , 2 , 3) New constraints C iab ≡ ∂ i ψ ab − Φ iab = 0 , C ijab ≡ 2 ∂ [ i Φ j ] ab = 0 To principal parts, obtain ∂ t ψ ab ≃ 0 , N k ∂ k Π ab − Ng ki ∂ k Φ iab + γ 2 N k ∂ k ψ ab , ∂ t Π ab ≃ N k ∂ k Φ iab − N ∂ i Π ab + N γ 2 ∂ i ψ ab , ∂ t Φ iab ≃ university-logo ( g ab = ψ ab + t a t b spatial metric, ( ∂ t ) a = Nt a + N a lapse & shift) Oliver Rinne (Caltech) GH Boundary Conditions: Stability&Accuracy GeoNum 11/21/2006 7 / 33

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