formulations of einstein equations and numerical...
TRANSCRIPT
![Page 1: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/1.jpg)
Formulations of Einstein
equations and numerical
solutions
Jerome Novak ([email protected])
Laboratoire Univers et Theories (LUTH)CNRS / Observatoire de Paris / Universite Paris-Diderot
based on collaboration withSilvano Bonazzola, Philippe Grandclement,
Eric Gourgoulhon & Nicolas Vasset
Institut de Physique Theorique, CEA Saclay, November,25th 2009
![Page 2: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/2.jpg)
Plan
1 Introduction
2 Formulations of Einstein equations
3 Spectral methods for numericalrelativity
4 Numerical simulation of black holes
![Page 3: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/3.jpg)
Plan
1 Introduction
2 Formulations of Einstein equations
3 Spectral methods for numericalrelativity
4 Numerical simulation of black holes
![Page 4: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/4.jpg)
Plan
1 Introduction
2 Formulations of Einstein equations
3 Spectral methods for numericalrelativity
4 Numerical simulation of black holes
![Page 5: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/5.jpg)
Plan
1 Introduction
2 Formulations of Einstein equations
3 Spectral methods for numericalrelativity
4 Numerical simulation of black holes
![Page 6: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/6.jpg)
Relativistic gravityIn general relativity (1915), space-time is afour-dimensional Lorentzian manifold, where gravitationalinteraction is described by the metric gµν .
Einstein equations
Rµν −1
2Rgµν = 8πTµν
They form a set of 10 second-order non-linear PDEs, withvery few (astro-)physically relevant exact solutions(Schwarzschild, Oppenheimer-Snyder, Kerr, . . . ).⇒approximate solutions:e.g. linearizing around the flat (Minkowski) solution invacuum gµν = ηµν + hµν :
(hµν −
1
2hηµν
)= −16πTµν .
![Page 7: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/7.jpg)
Relativistic gravityIn general relativity (1915), space-time is afour-dimensional Lorentzian manifold, where gravitationalinteraction is described by the metric gµν .
Einstein equations
Rµν −1
2Rgµν = 8πTµν
They form a set of 10 second-order non-linear PDEs, withvery few (astro-)physically relevant exact solutions(Schwarzschild, Oppenheimer-Snyder, Kerr, . . . ).⇒approximate solutions:e.g. linearizing around the flat (Minkowski) solution invacuum gµν = ηµν + hµν :
(hµν −
1
2hηµν
)= −16πTµν .
![Page 8: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/8.jpg)
Relativistic gravityIn general relativity (1915), space-time is afour-dimensional Lorentzian manifold, where gravitationalinteraction is described by the metric gµν .
Einstein equations
Rµν −1
2Rgµν = 8πTµν
They form a set of 10 second-order non-linear PDEs, withvery few (astro-)physically relevant exact solutions(Schwarzschild, Oppenheimer-Snyder, Kerr, . . . ).⇒approximate solutions:e.g. linearizing around the flat (Minkowski) solution invacuum gµν = ηµν + hµν :
(hµν −
1
2hηµν
)= −16πTµν .
![Page 9: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/9.jpg)
Relativistic gravityIn general relativity (1915), space-time is afour-dimensional Lorentzian manifold, where gravitationalinteraction is described by the metric gµν .
Einstein equations
Rµν −1
2Rgµν = 8πTµν
They form a set of 10 second-order non-linear PDEs, withvery few (astro-)physically relevant exact solutions(Schwarzschild, Oppenheimer-Snyder, Kerr, . . . ).⇒approximate solutions:e.g. linearizing around the flat (Minkowski) solution invacuum gµν = ηµν + hµν :
(hµν −
1
2hηµν
)= −16πTµν .
![Page 10: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/10.jpg)
Gravitational wavesAstrophysical sources
Using the linearized Einstein equations:
at first order h ∼ Q (mass quadrupole momentum of
the source), or further from the source h ∼ G
c4
E(`≥2)
r.
the total gravitational power of a source is
L ∼ G
c5s2ω6M2R4.
. . . introducing the Schwarzschild radius RS =2GM
c2and
ω = v/r:L ∼ c5
Gs2
(RS
R
)2 (vc
)6
⇒non-spherical, relativistic compact objects:
binary neutron stars or black holes,
supernovae and neutron star oscillations.
![Page 11: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/11.jpg)
Gravitational wavesAstrophysical sources
Using the linearized Einstein equations:
at first order h ∼ Q (mass quadrupole momentum of
the source), or further from the source h ∼ G
c4
E(`≥2)
r.
the total gravitational power of a source is
L ∼ G
c5s2ω6M2R4.
. . . introducing the Schwarzschild radius RS =2GM
c2and
ω = v/r:L ∼ c5
Gs2
(RS
R
)2 (vc
)6
⇒non-spherical, relativistic compact objects:
binary neutron stars or black holes,
supernovae and neutron star oscillations.
![Page 12: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/12.jpg)
Gravitational wavesAstrophysical sources
Using the linearized Einstein equations:
at first order h ∼ Q (mass quadrupole momentum of
the source), or further from the source h ∼ G
c4
E(`≥2)
r.
the total gravitational power of a source is
L ∼ G
c5s2ω6M2R4.
. . . introducing the Schwarzschild radius RS =2GM
c2and
ω = v/r:L ∼ c5
Gs2
(RS
R
)2 (vc
)6
⇒non-spherical, relativistic compact objects:
binary neutron stars or black holes,
supernovae and neutron star oscillations.
![Page 13: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/13.jpg)
Gravitational wavesDetectors
The effect of a wave ontwo tests-masses is thevariation of theirdistance ∆l/l ∼ h,measured by a LASERbeam.
LIGO: USA, Louisiana LIGO: USA, Washington
VIRGO: France/Italy (Pisa)Arms of these Michelson-type interfe-rometers are 3 km (VIRGO) and 4 km(LIGO) long . . . almost perfect vacuum.They are acquiring data since 2005,with a very complex data analysis⇒need for accurate wave patterns:perturbative and numericalapproaches.
![Page 14: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/14.jpg)
Gravitational wavesDetectors
The effect of a wave ontwo tests-masses is thevariation of theirdistance ∆l/l ∼ h,measured by a LASERbeam.
LIGO: USA, Louisiana LIGO: USA, Washington
VIRGO: France/Italy (Pisa)Arms of these Michelson-type interfe-rometers are 3 km (VIRGO) and 4 km(LIGO) long . . . almost perfect vacuum.They are acquiring data since 2005,with a very complex data analysis⇒need for accurate wave patterns:perturbative and numericalapproaches.
![Page 15: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/15.jpg)
A brief history of
numerical relativity1966 : May & White, Calculations of General-Relativistic
Collapse1975 : Butterworth & Ipser, Rapidly rotating fluid bodies in
general relativity1976 : Smarr, Cadez, DeWitt & Eppley, Collision of two black
holes1985 : Stark & Piran, Gravitational-Wave Emission from
Rotating Gravitational Collapse1993 : Abrahams & Evans, Vacuum axisymmetric gravitational
collapse1999 : Shibata, Fully general relativistic simulation of
coalescing binary neutron stars2005 : Pretorius, Evolution of Binary Black-Hole Spacetimes
![Page 16: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/16.jpg)
A brief history of
numerical relativity1966 : May & White, Calculations of General-Relativistic
Collapse1975 : Butterworth & Ipser, Rapidly rotating fluid bodies in
general relativity1976 : Smarr, Cadez, DeWitt & Eppley, Collision of two black
holes1985 : Stark & Piran, Gravitational-Wave Emission from
Rotating Gravitational Collapse1993 : Abrahams & Evans, Vacuum axisymmetric gravitational
collapse1999 : Shibata, Fully general relativistic simulation of
coalescing binary neutron stars2005 : Pretorius, Evolution of Binary Black-Hole Spacetimes
![Page 17: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/17.jpg)
A brief history of
numerical relativity1966 : May & White, Calculations of General-Relativistic
Collapse1975 : Butterworth & Ipser, Rapidly rotating fluid bodies in
general relativity1976 : Smarr, Cadez, DeWitt & Eppley, Collision of two black
holes1985 : Stark & Piran, Gravitational-Wave Emission from
Rotating Gravitational Collapse1993 : Abrahams & Evans, Vacuum axisymmetric gravitational
collapse1999 : Shibata, Fully general relativistic simulation of
coalescing binary neutron stars2005 : Pretorius, Evolution of Binary Black-Hole Spacetimes
![Page 18: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/18.jpg)
A brief history of
numerical relativity1966 : May & White, Calculations of General-Relativistic
Collapse1975 : Butterworth & Ipser, Rapidly rotating fluid bodies in
general relativity1976 : Smarr, Cadez, DeWitt & Eppley, Collision of two black
holes1985 : Stark & Piran, Gravitational-Wave Emission from
Rotating Gravitational Collapse1993 : Abrahams & Evans, Vacuum axisymmetric gravitational
collapse1999 : Shibata, Fully general relativistic simulation of
coalescing binary neutron stars2005 : Pretorius, Evolution of Binary Black-Hole Spacetimes
![Page 19: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/19.jpg)
A brief history of
numerical relativity1966 : May & White, Calculations of General-Relativistic
Collapse1975 : Butterworth & Ipser, Rapidly rotating fluid bodies in
general relativity1976 : Smarr, Cadez, DeWitt & Eppley, Collision of two black
holes1985 : Stark & Piran, Gravitational-Wave Emission from
Rotating Gravitational Collapse1993 : Abrahams & Evans, Vacuum axisymmetric gravitational
collapse1999 : Shibata, Fully general relativistic simulation of
coalescing binary neutron stars2005 : Pretorius, Evolution of Binary Black-Hole Spacetimes
![Page 20: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/20.jpg)
A brief history of
numerical relativity1966 : May & White, Calculations of General-Relativistic
Collapse1975 : Butterworth & Ipser, Rapidly rotating fluid bodies in
general relativity1976 : Smarr, Cadez, DeWitt & Eppley, Collision of two black
holes1985 : Stark & Piran, Gravitational-Wave Emission from
Rotating Gravitational Collapse1993 : Abrahams & Evans, Vacuum axisymmetric gravitational
collapse1999 : Shibata, Fully general relativistic simulation of
coalescing binary neutron stars2005 : Pretorius, Evolution of Binary Black-Hole Spacetimes
![Page 21: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/21.jpg)
A brief history of
numerical relativity1966 : May & White, Calculations of General-Relativistic
Collapse1975 : Butterworth & Ipser, Rapidly rotating fluid bodies in
general relativity1976 : Smarr, Cadez, DeWitt & Eppley, Collision of two black
holes1985 : Stark & Piran, Gravitational-Wave Emission from
Rotating Gravitational Collapse1993 : Abrahams & Evans, Vacuum axisymmetric gravitational
collapse1999 : Shibata, Fully general relativistic simulation of
coalescing binary neutron stars2005 : Pretorius, Evolution of Binary Black-Hole Spacetimes
![Page 22: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/22.jpg)
Formulations of Einstein equations
![Page 23: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/23.jpg)
Four-dimensional approachClassic approach in analytic studies: harmonic coordinatecondition, the coordinates xµµ=0...3 verify
xµ = 0.
⇒nice form of Einstein equations, with gαβ = Sαβ,⇒existence and uniqueness proofs in some cases.However, the gauge can be pathological (e.g. in presence ofmatter): necessity of some generalization for numericalimplementation.
xµ = Hµ,
with an arbitrary source. Generalized Harmonic gaugeChoice of Hµ ⇐⇒ choice of gauge
arbitrary function,evolution toward harmonic gauge ∂tHµ = −κ(t)Hµ,prescription from 3+1 formulations (see later).
first successful simulation of binary black hole evolution
![Page 24: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/24.jpg)
Four-dimensional approachClassic approach in analytic studies: harmonic coordinatecondition, the coordinates xµµ=0...3 verify
xµ = 0.
⇒nice form of Einstein equations, with gαβ = Sαβ,⇒existence and uniqueness proofs in some cases.However, the gauge can be pathological (e.g. in presence ofmatter): necessity of some generalization for numericalimplementation.
xµ = Hµ,
with an arbitrary source. Generalized Harmonic gaugeChoice of Hµ ⇐⇒ choice of gauge
arbitrary function,evolution toward harmonic gauge ∂tHµ = −κ(t)Hµ,prescription from 3+1 formulations (see later).
first successful simulation of binary black hole evolution
![Page 25: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/25.jpg)
Four-dimensional approachClassic approach in analytic studies: harmonic coordinatecondition, the coordinates xµµ=0...3 verify
xµ = 0.
⇒nice form of Einstein equations, with gαβ = Sαβ,⇒existence and uniqueness proofs in some cases.However, the gauge can be pathological (e.g. in presence ofmatter): necessity of some generalization for numericalimplementation.
xµ = Hµ,
with an arbitrary source. Generalized Harmonic gaugeChoice of Hµ ⇐⇒ choice of gauge
arbitrary function,evolution toward harmonic gauge ∂tHµ = −κ(t)Hµ,prescription from 3+1 formulations (see later).
first successful simulation of binary black hole evolution
![Page 26: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/26.jpg)
3+1 formalismDecomposition of spacetime and of Einstein equations
Evolution equations:∂Kij
∂t−LβKij =
−DiDjN +NRij − 2NKikKkj +
N [KKij + 4π((S − E)γij − 2Sij)]
Kij =1
2N
(∂γij
∂t+Diβj +Djβi
).
Constraint equations:
R+K2 −KijKij = 16πE,
DjKij −DiK = 8πJ i.
gµν dxµ dxν = −N2 dt2 + γij (dxi + βidt) (dxj + βjdt)
![Page 27: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/27.jpg)
3+1 formalismDecomposition of spacetime and of Einstein equations
Evolution equations:∂Kij
∂t−LβKij =
−DiDjN +NRij − 2NKikKkj +
N [KKij + 4π((S − E)γij − 2Sij)]
Kij =1
2N
(∂γij
∂t+Diβj +Djβi
).
Constraint equations:
R+K2 −KijKij = 16πE,
DjKij −DiK = 8πJ i.
gµν dxµ dxν = −N2 dt2 + γij (dxi + βidt) (dxj + βjdt)
![Page 28: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/28.jpg)
3+1 formalismDecomposition of spacetime and of Einstein equations
Evolution equations:∂Kij
∂t−LβKij =
−DiDjN +NRij − 2NKikKkj +
N [KKij + 4π((S − E)γij − 2Sij)]
Kij =1
2N
(∂γij
∂t+Diβj +Djβi
).
Constraint equations:
R+K2 −KijKij = 16πE,
DjKij −DiK = 8πJ i.
gµν dxµ dxν = −N2 dt2 + γij (dxi + βidt) (dxj + βjdt)
![Page 29: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/29.jpg)
Constrained / free
formulationsAs in electromagnetism, if the constraints are satisfiedinitially, they remain so for a solution of the evolutionequations.
free evolutionstart with initial data verifying the constraints,
solve only the 6 evolution equations,
recover a solution of all Einstein equations.
⇒apparition of constraint violating modes from round-offerrors. Considered cures:
Using of constraint damping terms and adapted gauges(many groups).Solving the constraints at every time-step(efficient elliptic solver?).
![Page 30: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/30.jpg)
Constrained / free
formulationsAs in electromagnetism, if the constraints are satisfiedinitially, they remain so for a solution of the evolutionequations.
free evolutionstart with initial data verifying the constraints,
solve only the 6 evolution equations,
recover a solution of all Einstein equations.
⇒apparition of constraint violating modes from round-offerrors. Considered cures:
Using of constraint damping terms and adapted gauges(many groups).Solving the constraints at every time-step(efficient elliptic solver?).
![Page 31: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/31.jpg)
Constrained / free
formulationsAs in electromagnetism, if the constraints are satisfiedinitially, they remain so for a solution of the evolutionequations.
free evolutionstart with initial data verifying the constraints,
solve only the 6 evolution equations,
recover a solution of all Einstein equations.
⇒apparition of constraint violating modes from round-offerrors. Considered cures:
Using of constraint damping terms and adapted gauges(many groups).Solving the constraints at every time-step(efficient elliptic solver?).
![Page 32: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/32.jpg)
Fully-constrained
formulation in Dirac gauge
Proposed by Bonazzola, Gourgoulhon, Grandclement & JN(2004): Define the conformal metric (carrying thedynamical degrees of freedom)
γij = Ψ4γij with Ψ =
(det γijdet fij
)1/12
,
choose the generalized Dirac gauge
∇(f)j γij = 0,
Then, one solves 4 constraint equations + 4 gauge equations(elliptic) at each time-step. Only 2 evolution equations.
![Page 33: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/33.jpg)
Fully-constrained
formulationProperties of the hyperbolic part
The hyperbolic part is obtained combining the evolutionequations:
∂Kij
∂t−LβKij = Sij and Kij =
1
2N
(∂γij
∂t+ . . .
),
to obtain a wave-type equation for γij.This system of evolution equations has been studied byCordero-Carrion et al. (2008):
the choice of Dirac gauge implies that the system isstrongly hyperbolic
can write it as conservation laws
no incoming characteristic in the case of black holeexcision technique
![Page 34: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/34.jpg)
Fully-constrained
formulationProperties of the hyperbolic part
The hyperbolic part is obtained combining the evolutionequations:
∂Kij
∂t−LβKij = Sij and Kij =
1
2N
(∂γij
∂t+ . . .
),
to obtain a wave-type equation for γij.This system of evolution equations has been studied byCordero-Carrion et al. (2008):
the choice of Dirac gauge implies that the system isstrongly hyperbolic
can write it as conservation laws
no incoming characteristic in the case of black holeexcision technique
![Page 35: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/35.jpg)
Elliptic partUniqueness issue
From the 4 constraints and the choice of time-slicing(gauge), an elliptic system of 5 non-linear equations can beformed
Elliptic part of Einstein equations, to be solved atevery time-stepWhen setting γij = f ij, the system reduces to theConformal-Flatness Condition (CFC).
Because of non-linear terms, theelliptic system may not converge⇒the case appears for dynamical,very compact matter and GWconfigurations (before appearanceof the black hole).
0.0 0.5 1.0 1.5 2.0
t
0.00
0.05
0.10
0.15
MA
DM
[ar
tbitr
ary
units
]
![Page 36: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/36.jpg)
A solution to the
uniqueness issueConsidering local uniqueness theorems for non-linearelliptic PDEs, it is possible to address the problem:⇒introducing auxiliary variables, to solve directly for themomentum constraints (Cordero-Carrion et al. (2009))
2nd fundamental form is rescaled bythe conformal factor Aij = Ψ10Kij,and decomposed into transverseand longitudinal parts ⇒solving foreach part:
longitudinal ⇐⇒ momentumconstraint,
transverse ⇐⇒ zero (CFC) orevolution.
![Page 37: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/37.jpg)
A solution to the
uniqueness issueConsidering local uniqueness theorems for non-linearelliptic PDEs, it is possible to address the problem:⇒introducing auxiliary variables, to solve directly for themomentum constraints (Cordero-Carrion et al. (2009))
2nd fundamental form is rescaled bythe conformal factor Aij = Ψ10Kij,and decomposed into transverseand longitudinal parts ⇒solving foreach part:
longitudinal ⇐⇒ momentumconstraint,
transverse ⇐⇒ zero (CFC) orevolution.
-50 -40 -30 -20 -10 0 10
t − tAH
10-3
10-2
10-1
100
Nc D1
D4SU
-50 -40 -30 -20 -10 0 101
2
3
4
5
ρ c/ρ
c,0
![Page 38: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/38.jpg)
Summary of Einstein
equationsconstrained scheme
evolution
∂Aij
∂t= ∇k∇kγ
ij + . . .
∂γij
∂t= 2NΨ−6Aij + . . .
with
det γij = 1,
∇(f)j γij = 0.
constraints
∇jAij = 8πΨ10Si,
∆Ψ = −2πΨ−1E
−Ψ−7AijAij8
,
∆NΨ = 2πNΨ−1 + . . .
withlimr→∞
γij = f ij, limr→∞
Ψ = limr→∞
N = 1.
![Page 39: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/39.jpg)
Spectral methods
for numerical relativity
![Page 40: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/40.jpg)
Simplified picture(see also Grandclement & JN 2009)
How to deal with functions on a computer?⇒a computer can manage only integersIn order to represent a function φ(x) (e.g. interpolate), onecan use:
a finite set of its values φii=0...N on a grid xii=0...N ,a finite set of its coefficients in a functional basisφ(x) '
∑Ni=0 ciΨi(x).
In order to manipulate a function (e.g. derive), eachapproach leads to:
finite differences schemes
φ′(xi) 'φ(xi+1)− φ(xi)
xi+1 − xispectral methods
φ′(x) 'N∑i=0
ciΨ′i(x)
![Page 41: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/41.jpg)
Simplified picture(see also Grandclement & JN 2009)
How to deal with functions on a computer?⇒a computer can manage only integersIn order to represent a function φ(x) (e.g. interpolate), onecan use:
a finite set of its values φii=0...N on a grid xii=0...N ,a finite set of its coefficients in a functional basisφ(x) '
∑Ni=0 ciΨi(x).
In order to manipulate a function (e.g. derive), eachapproach leads to:
finite differences schemes
φ′(xi) 'φ(xi+1)− φ(xi)
xi+1 − xispectral methods
φ′(x) 'N∑i=0
ciΨ′i(x)
![Page 42: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/42.jpg)
Simplified picture(see also Grandclement & JN 2009)
How to deal with functions on a computer?⇒a computer can manage only integersIn order to represent a function φ(x) (e.g. interpolate), onecan use:
a finite set of its values φii=0...N on a grid xii=0...N ,a finite set of its coefficients in a functional basisφ(x) '
∑Ni=0 ciΨi(x).
In order to manipulate a function (e.g. derive), eachapproach leads to:
finite differences schemes
φ′(xi) 'φ(xi+1)− φ(xi)
xi+1 − xispectral methods
φ′(x) 'N∑i=0
ciΨ′i(x)
![Page 43: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/43.jpg)
Convergence of Fourier
seriesφ(x) =
√1.5 + cos(x) + sin7 x
φ(x) 'N∑i=0
aiΨi(x) with Ψ2k = cos(kx), Ψ2k+1 = sin(kx)
![Page 44: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/44.jpg)
Convergence of Fourier
seriesφ(x) =
√1.5 + cos(x) + sin7 x
φ(x) 'N∑i=0
aiΨi(x) with Ψ2k = cos(kx), Ψ2k+1 = sin(kx)
![Page 45: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/45.jpg)
Convergence of Fourier
seriesφ(x) =
√1.5 + cos(x) + sin7 x
φ(x) 'N∑i=0
aiΨi(x) with Ψ2k = cos(kx), Ψ2k+1 = sin(kx)
![Page 46: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/46.jpg)
Convergence of Fourier
seriesφ(x) =
√1.5 + cos(x) + sin7 x
φ(x) 'N∑i=0
aiΨi(x) with Ψ2k = cos(kx), Ψ2k+1 = sin(kx)
![Page 47: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/47.jpg)
Convergence of Fourier
seriesφ(x) =
√1.5 + cos(x) + sin7 x
φ(x) 'N∑i=0
aiΨi(x) with Ψ2k = cos(kx), Ψ2k+1 = sin(kx)
![Page 48: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/48.jpg)
Use of orthogonal polynomials
The solutions (λi, ui)i∈N of a singular Sturm-Liouvilleproblem on the interval x ∈ [−1, 1]:
− (pu′)′+ qu = λwu,
with p > 0, C1, p(±1) = 0
are orthogonal with respect to the measure w:
(ui, uj) =
∫ 1
−1
ui(x)uj(x)w(x)dx = 0 for m 6= n,
form a spectral basis such that, if f(x) is smooth (C∞)
f(x) 'N∑i=0
ciui(x)
converges faster than any power of N (usually as e−N).
Gauss quadrature to compute the integrals giving the ci’s.Chebyshev, Legendre and, more generally any type ofJacobi polynomial enters this category.
![Page 49: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/49.jpg)
Use of orthogonal polynomials
The solutions (λi, ui)i∈N of a singular Sturm-Liouvilleproblem on the interval x ∈ [−1, 1]:
− (pu′)′+ qu = λwu,
with p > 0, C1, p(±1) = 0
are orthogonal with respect to the measure w:
(ui, uj) =
∫ 1
−1
ui(x)uj(x)w(x)dx = 0 for m 6= n,
form a spectral basis such that, if f(x) is smooth (C∞)
f(x) 'N∑i=0
ciui(x)
converges faster than any power of N (usually as e−N).
Gauss quadrature to compute the integrals giving the ci’s.Chebyshev, Legendre and, more generally any type ofJacobi polynomial enters this category.
![Page 50: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/50.jpg)
Use of orthogonal polynomials
The solutions (λi, ui)i∈N of a singular Sturm-Liouvilleproblem on the interval x ∈ [−1, 1]:
− (pu′)′+ qu = λwu,
with p > 0, C1, p(±1) = 0
are orthogonal with respect to the measure w:
(ui, uj) =
∫ 1
−1
ui(x)uj(x)w(x)dx = 0 for m 6= n,
form a spectral basis such that, if f(x) is smooth (C∞)
f(x) 'N∑i=0
ciui(x)
converges faster than any power of N (usually as e−N).
Gauss quadrature to compute the integrals giving the ci’s.Chebyshev, Legendre and, more generally any type ofJacobi polynomial enters this category.
![Page 51: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/51.jpg)
Use of orthogonal polynomials
The solutions (λi, ui)i∈N of a singular Sturm-Liouvilleproblem on the interval x ∈ [−1, 1]:
− (pu′)′+ qu = λwu,
with p > 0, C1, p(±1) = 0
are orthogonal with respect to the measure w:
(ui, uj) =
∫ 1
−1
ui(x)uj(x)w(x)dx = 0 for m 6= n,
form a spectral basis such that, if f(x) is smooth (C∞)
f(x) 'N∑i=0
ciui(x)
converges faster than any power of N (usually as e−N).
Gauss quadrature to compute the integrals giving the ci’s.Chebyshev, Legendre and, more generally any type ofJacobi polynomial enters this category.
![Page 52: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/52.jpg)
Method of weighted residuals
General form of an ODE of unknown u(x):
∀x ∈ [a, b], Lu(x) = s(x), and Bu(x)|x=a,b = 0,
The approximate solution is sought in the form
u(x) =N∑i=0
ciΨi(x).
The Ψii=0...N are called trial functions: they belong to afinite-dimension sub-space of some Hilbert space H[a,b].u is said to be a numerical solution if:
Bu = 0 for x = a, b,Ru = Lu− s is “small”.
Defining a set of test functions ξii=0...N and a scalarproduct on H[a,b], R is small iff:
∀i = 0 . . . N, (ξi, R) = 0.
It is expected that limN→∞ u = u, “true” solution of the ODE.
![Page 53: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/53.jpg)
Method of weighted residuals
General form of an ODE of unknown u(x):
∀x ∈ [a, b], Lu(x) = s(x), and Bu(x)|x=a,b = 0,
The approximate solution is sought in the form
u(x) =N∑i=0
ciΨi(x).
The Ψii=0...N are called trial functions: they belong to afinite-dimension sub-space of some Hilbert space H[a,b].u is said to be a numerical solution if:
Bu = 0 for x = a, b,Ru = Lu− s is “small”.
Defining a set of test functions ξii=0...N and a scalarproduct on H[a,b], R is small iff:
∀i = 0 . . . N, (ξi, R) = 0.
It is expected that limN→∞ u = u, “true” solution of the ODE.
![Page 54: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/54.jpg)
Method of weighted residuals
General form of an ODE of unknown u(x):
∀x ∈ [a, b], Lu(x) = s(x), and Bu(x)|x=a,b = 0,
The approximate solution is sought in the form
u(x) =N∑i=0
ciΨi(x).
The Ψii=0...N are called trial functions: they belong to afinite-dimension sub-space of some Hilbert space H[a,b].u is said to be a numerical solution if:
Bu = 0 for x = a, b,Ru = Lu− s is “small”.
Defining a set of test functions ξii=0...N and a scalarproduct on H[a,b], R is small iff:
∀i = 0 . . . N, (ξi, R) = 0.
It is expected that limN→∞ u = u, “true” solution of the ODE.
![Page 55: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/55.jpg)
Method of weighted residuals
General form of an ODE of unknown u(x):
∀x ∈ [a, b], Lu(x) = s(x), and Bu(x)|x=a,b = 0,
The approximate solution is sought in the form
u(x) =N∑i=0
ciΨi(x).
The Ψii=0...N are called trial functions: they belong to afinite-dimension sub-space of some Hilbert space H[a,b].u is said to be a numerical solution if:
Bu = 0 for x = a, b,Ru = Lu− s is “small”.
Defining a set of test functions ξii=0...N and a scalarproduct on H[a,b], R is small iff:
∀i = 0 . . . N, (ξi, R) = 0.
It is expected that limN→∞ u = u, “true” solution of the ODE.
![Page 56: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/56.jpg)
Various numerical methodstype of trial functions Ψ
finite-differences methods for local, overlapping polynomialsof low order,
finite-elements methods for local, smooth functions, whichare non-zero only on a sub-domain of [a, b],
spectral methods for global smooth functions on [a, b].
type of test functions ξ for spectral methods
tau method: ξi(x) = Ψi(x), but some of the test conditionsare replaced by the boundary conditions.
collocation method (pseudospectral): ξi(x) = δ(x− xi), atcollocation points. Some of the test conditions are replacedby the boundary conditions.
Galerkin method: the test and trial functions are chosento fulfill the boundary conditions.
![Page 57: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/57.jpg)
Various numerical methodstype of trial functions Ψ
finite-differences methods for local, overlapping polynomialsof low order,
finite-elements methods for local, smooth functions, whichare non-zero only on a sub-domain of [a, b],
spectral methods for global smooth functions on [a, b].
type of test functions ξ for spectral methods
tau method: ξi(x) = Ψi(x), but some of the test conditionsare replaced by the boundary conditions.
collocation method (pseudospectral): ξi(x) = δ(x− xi), atcollocation points. Some of the test conditions are replacedby the boundary conditions.
Galerkin method: the test and trial functions are chosento fulfill the boundary conditions.
![Page 58: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/58.jpg)
Inversion of linear ODEsThanks to the well-known recurrence relations of Legendreand Chebyshev polynomials, it is possible to express thecoefficients bii=0...N of
Lu(x) =N∑i=0
bi
∣∣∣∣ Pi(x)Ti(x)
, with u(x) =N∑i=0
ai
∣∣∣∣ Pi(x)Ti(x)
.
If L = d/dx, x×, . . . , and u(x) is represented by the vectoraii=0...N , L can be approximated by a matrix.
Resolution of a linear ODE
m
inversion of an (N + 1)× (N + 1) matrix
With non-trivial ODE kernels, one must add the boundaryconditions to the matrix to make it invertible!
![Page 59: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/59.jpg)
Inversion of linear ODEsThanks to the well-known recurrence relations of Legendreand Chebyshev polynomials, it is possible to express thecoefficients bii=0...N of
Lu(x) =N∑i=0
bi
∣∣∣∣ Pi(x)Ti(x)
, with u(x) =N∑i=0
ai
∣∣∣∣ Pi(x)Ti(x)
.
If L = d/dx, x×, . . . , and u(x) is represented by the vectoraii=0...N , L can be approximated by a matrix.
Resolution of a linear ODE
m
inversion of an (N + 1)× (N + 1) matrix
With non-trivial ODE kernels, one must add the boundaryconditions to the matrix to make it invertible!
![Page 60: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/60.jpg)
Inversion of linear ODEsThanks to the well-known recurrence relations of Legendreand Chebyshev polynomials, it is possible to express thecoefficients bii=0...N of
Lu(x) =N∑i=0
bi
∣∣∣∣ Pi(x)Ti(x)
, with u(x) =N∑i=0
ai
∣∣∣∣ Pi(x)Ti(x)
.
If L = d/dx, x×, . . . , and u(x) is represented by the vectoraii=0...N , L can be approximated by a matrix.
Resolution of a linear ODE
m
inversion of an (N + 1)× (N + 1) matrix
With non-trivial ODE kernels, one must add the boundaryconditions to the matrix to make it invertible!
![Page 61: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/61.jpg)
Some singular operators
u(x) 7→ u(x)
xis a linear operator, inverse of u(x) 7→ xu(x).
Its action on the coefficients aii=0...N representing theN -order approximation to a function u(x) can be computedas the product by a regular matrix.⇒The computation inthe coefficient space of u(x)/x, on the interval [−1, 1]always gives a finite result (both with Chebyshev andLegendre polynomials).⇒The actual operator which is thus computed is
u(x) 7→ u(x)− u(0)
x.
⇒Compute operators in spherical coordinates, withcoordinate singularities
e.g. ∆ =∂2
∂r2+
2
r
∂
∂r+
1
r2∆θϕ
![Page 62: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/62.jpg)
Some singular operators
u(x) 7→ u(x)
xis a linear operator, inverse of u(x) 7→ xu(x).
Its action on the coefficients aii=0...N representing theN -order approximation to a function u(x) can be computedas the product by a regular matrix.⇒The computation inthe coefficient space of u(x)/x, on the interval [−1, 1]always gives a finite result (both with Chebyshev andLegendre polynomials).⇒The actual operator which is thus computed is
u(x) 7→ u(x)− u(0)
x.
⇒Compute operators in spherical coordinates, withcoordinate singularities
e.g. ∆ =∂2
∂r2+
2
r
∂
∂r+
1
r2∆θϕ
![Page 63: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/63.jpg)
Some singular operators
u(x) 7→ u(x)
xis a linear operator, inverse of u(x) 7→ xu(x).
Its action on the coefficients aii=0...N representing theN -order approximation to a function u(x) can be computedas the product by a regular matrix.⇒The computation inthe coefficient space of u(x)/x, on the interval [−1, 1]always gives a finite result (both with Chebyshev andLegendre polynomials).⇒The actual operator which is thus computed is
u(x) 7→ u(x)− u(0)
x.
⇒Compute operators in spherical coordinates, withcoordinate singularities
e.g. ∆ =∂2
∂r2+
2
r
∂
∂r+
1
r2∆θϕ
![Page 64: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/64.jpg)
Some singular operators
u(x) 7→ u(x)
xis a linear operator, inverse of u(x) 7→ xu(x).
Its action on the coefficients aii=0...N representing theN -order approximation to a function u(x) can be computedas the product by a regular matrix.⇒The computation inthe coefficient space of u(x)/x, on the interval [−1, 1]always gives a finite result (both with Chebyshev andLegendre polynomials).⇒The actual operator which is thus computed is
u(x) 7→ u(x)− u(0)
x.
⇒Compute operators in spherical coordinates, withcoordinate singularities
e.g. ∆ =∂2
∂r2+
2
r
∂
∂r+
1
r2∆θϕ
![Page 65: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/65.jpg)
Time discretizationFormally, the representation (and manipulation) of f(t) isthe same as that of f(x).⇒in principle, one should be able to represent a functionu(x, t) and solve time-dependent PDEs only using spectralmethods...but this is not the way it is done! Two works:
Ierley et al. (1992): study of the Korteweg de Vriesand Burger equations, Fourier in space and Chebyshevin time ⇒time-stepping restriction.Hennig and Ansorg (2008): study of non-linear (1+1)wave equation, with conformal compactification inMinkowski space-time. ⇒nice spectral convergence.
WHY?
poor a priori knowledge of the exact time interval,too big matrices for full 3+1 operators (∼ 304 × 304),finite-differences time-stepping errors can be quitesmall.
![Page 66: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/66.jpg)
Time discretizationFormally, the representation (and manipulation) of f(t) isthe same as that of f(x).⇒in principle, one should be able to represent a functionu(x, t) and solve time-dependent PDEs only using spectralmethods...but this is not the way it is done! Two works:
Ierley et al. (1992): study of the Korteweg de Vriesand Burger equations, Fourier in space and Chebyshevin time ⇒time-stepping restriction.Hennig and Ansorg (2008): study of non-linear (1+1)wave equation, with conformal compactification inMinkowski space-time. ⇒nice spectral convergence.
WHY?
poor a priori knowledge of the exact time interval,too big matrices for full 3+1 operators (∼ 304 × 304),finite-differences time-stepping errors can be quitesmall.
![Page 67: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/67.jpg)
Time discretizationFormally, the representation (and manipulation) of f(t) isthe same as that of f(x).⇒in principle, one should be able to represent a functionu(x, t) and solve time-dependent PDEs only using spectralmethods...but this is not the way it is done! Two works:
Ierley et al. (1992): study of the Korteweg de Vriesand Burger equations, Fourier in space and Chebyshevin time ⇒time-stepping restriction.Hennig and Ansorg (2008): study of non-linear (1+1)wave equation, with conformal compactification inMinkowski space-time. ⇒nice spectral convergence.
WHY?
poor a priori knowledge of the exact time interval,too big matrices for full 3+1 operators (∼ 304 × 304),finite-differences time-stepping errors can be quitesmall.
![Page 68: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/68.jpg)
Explicit / implicit schemesLet us look for the numerical solution of (L acts only on x):
∀t ≥ 0, ∀x ∈ [−1, 1],∂u(x, t)
∂t= Lu(x, t),
with good boundary conditions. Then, with δt thetime-step: ∀J ∈ N, uJ(x) = u(x, J × δt), it is possible todiscretize the PDE as
uJ+1(x) = uJ(x) + δt LuJ(x): explicit time scheme(forward Euler); easy to implement, fast but limited bythe CFL condition.
uJ+1(x)− δt LuJ+1(x) = uJ(x): implicit time scheme(backward Euler); one must solve an equation (ODE)to get uJ+1, the matrix approximating it here isI − δt L. Allows longer time-steps but slower andlimited to second-order schemes.
![Page 69: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/69.jpg)
Explicit / implicit schemesLet us look for the numerical solution of (L acts only on x):
∀t ≥ 0, ∀x ∈ [−1, 1],∂u(x, t)
∂t= Lu(x, t),
with good boundary conditions. Then, with δt thetime-step: ∀J ∈ N, uJ(x) = u(x, J × δt), it is possible todiscretize the PDE as
uJ+1(x) = uJ(x) + δt LuJ(x): explicit time scheme(forward Euler); easy to implement, fast but limited bythe CFL condition.
uJ+1(x)− δt LuJ+1(x) = uJ(x): implicit time scheme(backward Euler); one must solve an equation (ODE)to get uJ+1, the matrix approximating it here isI − δt L. Allows longer time-steps but slower andlimited to second-order schemes.
![Page 70: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/70.jpg)
Explicit / implicit schemesLet us look for the numerical solution of (L acts only on x):
∀t ≥ 0, ∀x ∈ [−1, 1],∂u(x, t)
∂t= Lu(x, t),
with good boundary conditions. Then, with δt thetime-step: ∀J ∈ N, uJ(x) = u(x, J × δt), it is possible todiscretize the PDE as
uJ+1(x) = uJ(x) + δt LuJ(x): explicit time scheme(forward Euler); easy to implement, fast but limited bythe CFL condition.
uJ+1(x)− δt LuJ+1(x) = uJ(x): implicit time scheme(backward Euler); one must solve an equation (ODE)to get uJ+1, the matrix approximating it here isI − δt L. Allows longer time-steps but slower andlimited to second-order schemes.
![Page 71: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/71.jpg)
Multi-domain approachMulti-domain technique : several touching, or overlapping,domains (intervals), each one mapped on [−1, 1].
boundary between two domains can be the place of adiscontinuity ⇒recover spectral convergence,one can set a domain with more coefficients(collocation points) in a region where much resolutionis needed ⇒fixed mesh refinement,2D or 3D, allows to build a complex domain fromseveral simpler ones,
Depending on the PDE, matching conditions are imposedat y = y0 ⇐⇒ boundary conditions in each domain.
![Page 72: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/72.jpg)
Multi-domain approachMulti-domain technique : several touching, or overlapping,domains (intervals), each one mapped on [−1, 1].
boundary between two domains can be the place of adiscontinuity ⇒recover spectral convergence,one can set a domain with more coefficients(collocation points) in a region where much resolutionis needed ⇒fixed mesh refinement,2D or 3D, allows to build a complex domain fromseveral simpler ones,
Depending on the PDE, matching conditions are imposedat y = y0 ⇐⇒ boundary conditions in each domain.
![Page 73: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/73.jpg)
Multi-domain approachMulti-domain technique : several touching, or overlapping,domains (intervals), each one mapped on [−1, 1].
boundary between two domains can be the place of adiscontinuity ⇒recover spectral convergence,one can set a domain with more coefficients(collocation points) in a region where much resolutionis needed ⇒fixed mesh refinement,2D or 3D, allows to build a complex domain fromseveral simpler ones,
Depending on the PDE, matching conditions are imposedat y = y0 ⇐⇒ boundary conditions in each domain.
![Page 74: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/74.jpg)
Multi-domain approachMulti-domain technique : several touching, or overlapping,domains (intervals), each one mapped on [−1, 1].
boundary between two domains can be the place of adiscontinuity ⇒recover spectral convergence,one can set a domain with more coefficients(collocation points) in a region where much resolutionis needed ⇒fixed mesh refinement,2D or 3D, allows to build a complex domain fromseveral simpler ones,
Domain 1 Domain 2x
1=-1 x
1=1 x
2=-1 x
2=1
y=ay = y
0
y=b
Depending on the PDE, matching conditions are imposedat y = y0 ⇐⇒ boundary conditions in each domain.
![Page 75: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/75.jpg)
Mappings and multi-DIn two spatial dimensions, the usualtechnique is to write a function as:
f : Ω = [−1, 1]× [−1, 1]→ R
f(x, y) =Nx∑i=0
Ny∑j=0
cijPi(x)Pj(y)
ΠΩΩ
The domain Ω is then mapped to the real physical domain,trough some mapping Π : (x, y) 7→ (X,Y ) ∈ Ω.⇒When computing derivatives, the Jacobian of Π is used.
compactificationA very convenient mapping in spherical coordinates is
x ∈ [−1, 1] 7→ r =1
α(x− 1),
to impose boundary condition for r →∞ at x = 1.
![Page 76: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/76.jpg)
Mappings and multi-DIn two spatial dimensions, the usualtechnique is to write a function as:
f : Ω = [−1, 1]× [−1, 1]→ R
f(x, y) =Nx∑i=0
Ny∑j=0
cijPi(x)Pj(y)
ΠΩΩ
The domain Ω is then mapped to the real physical domain,trough some mapping Π : (x, y) 7→ (X,Y ) ∈ Ω.⇒When computing derivatives, the Jacobian of Π is used.
compactificationA very convenient mapping in spherical coordinates is
x ∈ [−1, 1] 7→ r =1
α(x− 1),
to impose boundary condition for r →∞ at x = 1.
![Page 77: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/77.jpg)
Example:3D Poisson equation, with non-compact support
To solve ∆φ(r, θ, ϕ) = s(r, θ, ϕ), with s extending to infinity.
Nucleusr = αξ, 0 ≤ ξ ≤ 1
T2i
(ξ) for l even
T2i+1
(ξ) for l odd
Compactified domain
r = 1
β(ξ − 1), 0 ≤ ξ ≤ 1
T_i(ξ)
setup two domains in the radialdirection: one to deal with thesingularity at r = 0, the otherwith a compactified mapping.
In each domain decompose theangular part of both fields ontospherical harmonics:
φ(ξ, θ, ϕ) '`max∑`=0
m=∑m=−`
φ`m(ξ)Y m` (θ, ϕ),
∀(`,m) solve the ODE:d2φ`m
dξ2+
2ξ
dφ`mdξ− `(`+ 1)φ`m
ξ2= s`m(ξ),
match between domains, with regularity conditions atr = 0, and boundary conditions at r →∞.
![Page 78: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/78.jpg)
Example:3D Poisson equation, with non-compact support
To solve ∆φ(r, θ, ϕ) = s(r, θ, ϕ), with s extending to infinity.
Nucleusr = αξ, 0 ≤ ξ ≤ 1
T2i
(ξ) for l even
T2i+1
(ξ) for l odd
Compactified domain
r = 1
β(ξ − 1), 0 ≤ ξ ≤ 1
T_i(ξ)
setup two domains in the radialdirection: one to deal with thesingularity at r = 0, the otherwith a compactified mapping.
In each domain decompose theangular part of both fields ontospherical harmonics:
φ(ξ, θ, ϕ) '`max∑`=0
m=∑m=−`
φ`m(ξ)Y m` (θ, ϕ),
∀(`,m) solve the ODE:d2φ`m
dξ2+
2ξ
dφ`mdξ− `(`+ 1)φ`m
ξ2= s`m(ξ),
match between domains, with regularity conditions atr = 0, and boundary conditions at r →∞.
![Page 79: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/79.jpg)
Numerical simulation of black holes
![Page 80: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/80.jpg)
Puncture methods. . . it is not yet clear how and why they work. Hannam et al. (2007)
black holes are described in the initial data incoordinates that do not reach the physical singularity,
⇒ the coordinates follow a wormhole through anothercopy of the asymptotically flat exterior spacetime,this is compactified so that infinity is represented by asingle point, called “puncture”.
γij = Ψ4γij with Ψ ∼ 1
r, use of φ = log Ψ or χ = Ψ−4.
BUTDuring the evolution the time-slice loses contact with thesecond asymptotically flat end, and finishes on a cylinder offinite radius.
Ψ(t = 0) = O(
1
r
)evolves into Ψ(t > 0) = O
(1√r
).
Use of the shift vector βi to generate motion.
![Page 81: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/81.jpg)
Puncture methods. . . it is not yet clear how and why they work. Hannam et al. (2007)
black holes are described in the initial data incoordinates that do not reach the physical singularity,
⇒ the coordinates follow a wormhole through anothercopy of the asymptotically flat exterior spacetime,this is compactified so that infinity is represented by asingle point, called “puncture”.
γij = Ψ4γij with Ψ ∼ 1
r, use of φ = log Ψ or χ = Ψ−4.
BUTDuring the evolution the time-slice loses contact with thesecond asymptotically flat end, and finishes on a cylinder offinite radius.
Ψ(t = 0) = O(
1
r
)evolves into Ψ(t > 0) = O
(1√r
).
Use of the shift vector βi to generate motion.
![Page 82: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/82.jpg)
Puncture methods. . . it is not yet clear how and why they work. Hannam et al. (2007)
black holes are described in the initial data incoordinates that do not reach the physical singularity,
⇒ the coordinates follow a wormhole through anothercopy of the asymptotically flat exterior spacetime,this is compactified so that infinity is represented by asingle point, called “puncture”.
γij = Ψ4γij with Ψ ∼ 1
r, use of φ = log Ψ or χ = Ψ−4.
BUTDuring the evolution the time-slice loses contact with thesecond asymptotically flat end, and finishes on a cylinder offinite radius.
Ψ(t = 0) = O(
1
r
)evolves into Ψ(t > 0) = O
(1√r
).
Use of the shift vector βi to generate motion.
![Page 83: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/83.jpg)
Excision techniquesapparent horizons as a boundary
Remove a neighborhood of the central singularity fromcomputational domain;
Replace it with boundary conditions on this newly obtainedboundary (usually, a sphere),
Until now, imposition of apparent horizon / isolated horizonproperties: zero expansion of outgoing light rays.
⇒New views on the concept of black hole,following works by Hayward, Ashtekar andKrishnan:
Quasi-local approach, making theblack hole a causal object;
For hydrodynamic, electromagneticand gravitational waves (Diracgauge): no incoming characteristics.
![Page 84: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/84.jpg)
Excision techniqueKerr solution from boundary conditions
Can one recover a Kerr black hole only from boundaryconditions and Einstein equations?⇒Many computations with CFC, but there is no timeslicing in which (the spatial part of) Kerr solution can beconformally flat (Garat & Price 2000).Vasset, JN & Jaramillo (2009) recover full Kerr solution
constant value (N), zero expansion on the horizon (ψ);
rotation state for βθ, βφ and isolated horizon for βr;
NO condition for γij;
+ asymptotic flatness and Einstein equations!
In particular, no symmetry requirement has been imposedin the “bulk” (only on the horizon) ⇒illustration of therigidity theorem by Hawking & Ellis (1973).
![Page 85: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/85.jpg)
Excision techniqueKerr solution from boundary conditions
Can one recover a Kerr black hole only from boundaryconditions and Einstein equations?⇒Many computations with CFC, but there is no timeslicing in which (the spatial part of) Kerr solution can beconformally flat (Garat & Price 2000).Vasset, JN & Jaramillo (2009) recover full Kerr solution
constant value (N), zero expansion on the horizon (ψ);
rotation state for βθ, βφ and isolated horizon for βr;
NO condition for γij;
+ asymptotic flatness and Einstein equations!
In particular, no symmetry requirement has been imposedin the “bulk” (only on the horizon) ⇒illustration of therigidity theorem by Hawking & Ellis (1973).
![Page 86: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/86.jpg)
Excision techniqueKerr solution from boundary conditions
Can one recover a Kerr black hole only from boundaryconditions and Einstein equations?⇒Many computations with CFC, but there is no timeslicing in which (the spatial part of) Kerr solution can beconformally flat (Garat & Price 2000).Vasset, JN & Jaramillo (2009) recover full Kerr solution
constant value (N), zero expansion on the horizon (ψ);
rotation state for βθ, βφ and isolated horizon for βr;
NO condition for γij;
+ asymptotic flatness and Einstein equations!
In particular, no symmetry requirement has been imposedin the “bulk” (only on the horizon) ⇒illustration of therigidity theorem by Hawking & Ellis (1973).
![Page 87: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/87.jpg)
Summary - PerspectivesMany new results in numerical relativity,The Fully-constrained Formulation is needed forlong-term evolutions, particularly in the cases ofgravitational collapse,This formulation is now well-studied and stable.
Many of the numerical features presented here are availablein the lorene library: http://lorene.obspm.fr, publiclyavailable under GPL.Future directions:
Implementation of FCF and excision methods in thecollapse code to simulate the formation of a black hole;Use of excision techniques in the dynamical case
⇒most of groups are now heading toward more complexphysics: electromagnetic field, realistic equation of state formatter, . . .
![Page 88: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/88.jpg)
Summary - PerspectivesMany new results in numerical relativity,The Fully-constrained Formulation is needed forlong-term evolutions, particularly in the cases ofgravitational collapse,This formulation is now well-studied and stable.
Many of the numerical features presented here are availablein the lorene library: http://lorene.obspm.fr, publiclyavailable under GPL.Future directions:
Implementation of FCF and excision methods in thecollapse code to simulate the formation of a black hole;Use of excision techniques in the dynamical case
⇒most of groups are now heading toward more complexphysics: electromagnetic field, realistic equation of state formatter, . . .
![Page 89: Formulations of Einstein equations and numerical solutionsluthier/novak/presentations/ipht_09.pdfEinstein equations R 1 2 Rg = 8ˇT They form a set of 10 second-order non-linear PDEs,](https://reader033.vdocuments.us/reader033/viewer/2022050612/5fb2eef523e95d03bb3956d8/html5/thumbnails/89.jpg)
Summary - PerspectivesMany new results in numerical relativity,The Fully-constrained Formulation is needed forlong-term evolutions, particularly in the cases ofgravitational collapse,This formulation is now well-studied and stable.
Many of the numerical features presented here are availablein the lorene library: http://lorene.obspm.fr, publiclyavailable under GPL.Future directions:
Implementation of FCF and excision methods in thecollapse code to simulate the formation of a black hole;Use of excision techniques in the dynamical case
⇒most of groups are now heading toward more complexphysics: electromagnetic field, realistic equation of state formatter, . . .