Introduction to Numerical Relativity II Numerics,...

Introduction to Numerical Relativity IINumerics, Codes and Applications

S. Husa

University of the Balearic Islands, [email protected]

August 3, 2010

S. Husa (UIB) Numerical Relativity Vienna 1 / 26

Outline

1 Numerical stability and Well-Posedness

2 Computational Infrastructure

3 ApplicationsCritical CollapseBinary Black Hole DynamicsInterlude: GW detectionUltrarelativistic Collisions

4 Getting started


Outline




4 Getting started


Outline




4 Getting started


Outline




4 Getting started


Numerical stability and Well-Posedness

NR at the turn of the century

90’s: success with critical collapse in1+1, but still little success with BBHsafter > 3 decades.

Lnhab = 2Kab

LnKab = −α−1DaDbα

+Rab + KKab − 2KacK cb

York-ADM eqs. geometrically elegant &minimalistic, but initial value problemwell posed only in spherical symmetry.

Pair geometric with understanding thedark side – EEs as PDEs!





Lnhab = 2Kab









Lnhab = 2Kab







Conditioning as a technical starting point

Model problem F (x , y) = 0 – how sensitive is the dependence y(x)?

Condition number: worst possible effect on y when x is perturbed

Consider perturbed eq. F (x + δx , y + δy) = 0,define

K = supδx

||δy ||/||y ||||δx ||/||x ||

K small: well conditioned, K large: ill conditioned, K (y) =∞: ill-posed, unstable



Well–posedness and Stability

WP: A unique solution exists at least for some time (when gauge is chosen),depends continuously on initial data.

||u(t)|| ≤ Kea t ||u(0)||,

Exponential term ensures robustness w.r.t. lower order terms!

For ill–posed problems a,K depend on ID: Higher frequencies correspond to largera,K ⇒ better resolution → worse solution.

WP (stable) in numerical context for iterative problem (eαtn ok, eαn not):

vn+1 = Q(tn, vn, vn+1)vn : ||vn|| ≤ Keαtn ||v 0|| ∀v 0

Lax equivalence theorem:“a consistent (formally convergent) finite difference scheme for a linear PDE forwhich the initial value problem is well posed is convergent iff it is stable.”



Example: ODEs

ODE IVPs are well–posed, solutions may blow up in finite time:

y ′ = λy 2, y(0) = y0 → y(x) = y0/(x y0− 1).

To analyze numerical stability, perform Von Neumann analysis and consider

y ′ = λy , y(0) = y0,

solve numerically with forward Euler algorithm

yn+1 = yn + hy ′n = yn + hλyn → |yn+1|/|yn| = |1 + hλ|

– unstable for h > −2/λ

For λ > 0 even a stable algorithm will suffer from ill-conditioning.



“Evolution problem” & “well-posedness wars”

Ad–hoc methods for first order in time/second order in space systems dominating“astrophysical applications” vs. standard theory of first order systems!⇒ much confusion about what the essential problems are!

How to solve ODEs? → convert to first order, choose “good” variables, a stabletime integrator and appropriate time step, special treatment for singularities etc.!

PDEs:Reduction to first order in time → (FTSS) new evolution equations

Reduction to first order in space → new evolution & constraint equations!

Reduction to first order in space changes solution space!

FTSS formulation is convenient for “technical reasons”: expect higher accuracy,less constraints, easier to analyze (WP & numerically stable not enough!) . . .



First Order Systems in a Nutshell

Consider a linear first order system with constant coefficients: u = A∂xu + l .o.t.

Solve in Fourier space: ˙u = IωAu → u = e IωtAu0. (complex eigenvalues andJordan Blocks create trouble!)

Real speeds: (weakly) hyperbolic, ∃ norm → well posed in absence of l.o.t.

Complete set of eigenvectors (characteristic variables span solution space):well posed (strongly hyperbolic)

strongly hyperbolic + conserved energy: symmetrizable

Method of Lines: split time and space discretizations, leave time continous at first→ PDE reduces to family of coupled ODEs.

Semidiscrete analogous! Fully discrete: semidiscrete stability + Von-Neumann.

Treating ODEs in Fourier space puts first and second order in space on same level!



Example: ADM 1D, “densitized lapse”

hii = 2Kii

Kxx = 12∂xxhxx + ∂xx (hyy + hzz)

Kjj = 12∂xxhjj (j = y , z)

All characteristic speeds real; Jordan normal form of first order system:

−1 0 0 0 0 00 −1 1 0 0 00 0 −1 0 0 00 0 0 1 0 00 0 0 0 1 10 0 0 0 0 1

Decoupling the xx components from the yy and zz components leads towell–posed systems and good numerical results.

Direct Fourier-analysis of FTSS problem quickly leads to same conclusion: ADMill-posed!



“Generalized Harmonic”

Rµν = −1

2gλρ gµν,λρ +∇(µΓν) + Γλ

ηµ gηδ gλρ Γρ

δν + 2 Γδ

λη gδρ gλ(µ Γν)

ηρ.

Consider Γν as given function, 2xµ = Γµ.

∂t α− α,a βa = α2 (K − nνΓν),

∂t βa − βa

,b βb = α2 (γa − Da log α− ha

ν Γν),

C a ≡ Γµ −2xµ, Cµ = 0⇒ Einstein constraints = 0!

Important: second order in time equations for lapse and shift in combination withharmonic decomposition!

4-dimensional form of metric plays no role! – 3+1 decomposition can be used thesame way! Broad choice of variables!

Harmonic evolution appears prone to constraint violating modes, larger number ofdamping terms have been developed.

Harmonic lapse leads to evolutions where the spatial slice approaches a BHsingularity for large times.



BSSN: the workhorse formulation for BHs and astrophysics

Core idea: York-ADM + factor our spatial conformal factor + introduce 1 extravariable to achieve hyperbolicity – follow KISS principle.

ϕ = (1/12) log(detγij)γij = e−4ϕγijK = γ ijKij ,

Aij = e−4ϕ(Kij − (1/3)γijK ),

Γi = Γijk γ

jk

Making a standard choice for adding constraints (several “natural” ambiguities inthis system – not yet completely analyzed!) we get

Lnϕ = −(1/6)αK ,

Lnγij = −2αAij ,

LnK = −D iDiα + αAij Aij + (1/3)αK 2,

LnAij = −e−4ϕ(DiDjα)TF + e−4ϕα(Rij)TF + αK Aij − 2αAik Ak

j ,

LnΓi = −2(∂jα)Aij + 2α(Γijk Akj − (2/3)γ ij(∂jK ) + 6Aij(∂jϕ)

).


Computational Infrastructure

Code and Computational resources

Scaling of resources in N space dimensions:

∆t → ∆t/2, ∆x → ∆x/2

⇒ Memory→ 2N ×Memory, Operations→ 2N+1 × Ops., Error→ 2−p × Err.

for p-th order scheme.

E.g. 3 space dimensions: break-even at fourth order accuracy. BBH work requiresat least 6th order for satisfactory phase error.

Typical memory requirements of 3D BBH codes:> 100 grid functions, > 1003 boxes in 10 refinement levels: ≈ 100GByte.

Runs take several days to weeks → BBH groups use several million CPUhours/year.

European groups seriously starved for computer time, hope for PRACE.

What is the right level of cooperation/competition in exploring the parameterspace? How are similar challenges handled in other communities?



Computational infrastructure for NR

Einstein toolkit, http://einsteintoolkit.org, collection of open sourcecode for NR, largely build on Cactus framework for parallelisation(MPI/OpenMP based) – http://www.cactuscode.org.

Simulation Factory: compiling and deploying applications on machines andmanaging simulations.Kranc mathematica package to generate codes from high-level descriptions ofPDE problems.Visualization tools.Alpaca tools for performance measuring and code testing.Full-featured GR production code including BH & NS initial data, evolutioncode with mesh-refinement, AH & EH finders, GW extraction, . . .Full checkpointing support.Private multipatch and characteristic codes available.

www.ApplesWithApples.org – suite of standardized testbeds for NR

xAct suite of Mathematica packages, http://www.xact.es/ (J. M. MartinGarcia).





Simulation Factory: compiling and deploying applications on machines andmanaging simulations.

Kranc mathematica package to generate codes from high-level descriptions ofPDE problems.Visualization tools.Alpaca tools for performance measuring and code testing.Full-featured GR production code including BH & NS initial data, evolutioncode with mesh-refinement, AH & EH finders, GW extraction, . . .Full checkpointing support.Private multipatch and characteristic codes available.







Simulation Factory: compiling and deploying applications on machines andmanaging simulations.Kranc mathematica package to generate codes from high-level descriptions ofPDE problems.

Visualization tools.Alpaca tools for performance measuring and code testing.Full-featured GR production code including BH & NS initial data, evolutioncode with mesh-refinement, AH & EH finders, GW extraction, . . .Full checkpointing support.Private multipatch and characteristic codes available.







Simulation Factory: compiling and deploying applications on machines andmanaging simulations.Kranc mathematica package to generate codes from high-level descriptions ofPDE problems.Visualization tools.

Alpaca tools for performance measuring and code testing.Full-featured GR production code including BH & NS initial data, evolutioncode with mesh-refinement, AH & EH finders, GW extraction, . . .Full checkpointing support.Private multipatch and characteristic codes available.







Simulation Factory: compiling and deploying applications on machines andmanaging simulations.Kranc mathematica package to generate codes from high-level descriptions ofPDE problems.Visualization tools.Alpaca tools for performance measuring and code testing.

Full-featured GR production code including BH & NS initial data, evolutioncode with mesh-refinement, AH & EH finders, GW extraction, . . .Full checkpointing support.Private multipatch and characteristic codes available.







Simulation Factory: compiling and deploying applications on machines andmanaging simulations.Kranc mathematica package to generate codes from high-level descriptions ofPDE problems.Visualization tools.Alpaca tools for performance measuring and code testing.Full-featured GR production code including BH & NS initial data, evolutioncode with mesh-refinement, AH & EH finders, GW extraction, . . .

Full checkpointing support.Private multipatch and characteristic codes available.







Simulation Factory: compiling and deploying applications on machines andmanaging simulations.Kranc mathematica package to generate codes from high-level descriptions ofPDE problems.Visualization tools.Alpaca tools for performance measuring and code testing.Full-featured GR production code including BH & NS initial data, evolutioncode with mesh-refinement, AH & EH finders, GW extraction, . . .Full checkpointing support.

Private multipatch and characteristic codes available.


























Why mesh refinement?

Many numerical relativity problems require adaptive mesh refinement:

Critical collapse at the threshold of BH-formation (self-similar solutions)

binary black holes:

many different length scales:

the individual black holes (radiation efficiency ∼ compactness of source)

orbital scale

wave scale > 10M

1/rn background falloff

Binary black hole simulations do not require full AMR: refinement boxes justneed to track BHs!

Variant of Berger-Oliger (fine grids evolve faster) + extra buffers


Applications Critical Collapse

First Success Story: Critical Collapse

Possible final states of gravitating systems?Can naked singularities form out of “regular” initial data?Can we create arbitrarily small BHs – arbitrarily large curvature?

Sufficiently “strong” data contain trapped surfaces⇒ singularitiesSufficiently “weak” data decay to Minkowski.

Choptuik ≈ 1990: Numerical “zoom” (bisection) on the transition between“decay” to BH-formation in the space of spherical scalar field initial data.

Critical data correspond to ahypersurface of codimension 1 in thespace of initial data

Universal critical solution is discretelyself-similar, corresponds to intermediateattractor in the language of dynamicalsystems. Mass scaling MBH ∝ |p−p?|γ .

Flat space fixed point

Black hole fixed point

thresholdBlack hole

pointCritical

p = p*

initial data

family of

1−parameter

p < p*

p > p*



First Success Story: Critical Collapse

Possible final states of gravitating systems?

Can naked singularities form out of “regular” initial data?

Can we create arbitrarily small BHs – arbitrarily large curvature?

Sufficiently “strong” data contain trapped surfaces⇒ singularities

Sufficiently “weak” data decay to Minkowski.

Choptuik ≈ 1990: Numerical “zoom” (bisection) on the transition between“decay” to BH-formation in the space of spherical scalar field initial data.

Critical data correspond to ahypersurface of codimension 1 in thespace of initial data

Universal critical solution is discretelyself-similar, corresponds to intermediateattractor in the language of dynamicalsystems. Mass scaling MBH ∝ |p−p?|γ .



Critical Collapse II

Since Choptuik’s initial discovery critical behaviour has been found for a numberof fields (complex scalar fields, σ-models, Yang-Mills, fluids, . . . ).

Most work in spherical symmetry, but also in 2+1, and for zoom-whirl BBH orbitsin neutron star collisions in 3+1.

Critical collapse for pure gravitational waves has not been confirmed.

Spherical symmetry: instead of sophisticated AMR, double-null coordinates can beused, focusing of null-geodesics concentrates grid-points in regions of largecurvature.

Attractors show symmetries: exact or discrete self-similarity, stationarity ortime-periodic solutions (breathers).



Critical Collapse III

Two principal types of critial behaviour found:

Type II: analogous to second order phase transitions in statistical mechanics:

MBH ∝ |p − p?|γ .

Associated with (discretely) self-simlar solutions with 1 unstable mode.

Type I: analogous to first order phase transitions in statistical mechanics:critical black hole mass is finite.Associated with stationary/periodic solutions with 1 unstable mode.

Many systems, such as the massive scalar field, show both type I and type IIcritical phenomena, in different regions of the space of initial data.

Can critical phenomena be observed in nature (natural fine-tuning)?

”Critical Phenomena in Gravitational Collapse” Carsten Gundlach and Jos M.Martn-Garca http://www.livingreviews.org/lrr-2007-5


Applications Binary Black Hole Dynamics

Binary Black Hole Problem

Standard technique of math. relativityto treat black hole initial data: modelBH as nontrivial topology – add(compactified) infinities – “punctures”![Beig & O Murchadha, CQG (1994)]

Spacelike infinities enforce minimalsurfaces → apparent horizons → BHs.

Staying away from the singularity:

Excision technique: do not evolve values inside a pure outflow boundary

Singularity avoiding slicings: Choose coordinates such that the physicalsingularity is approached only at infinite coordinate time – or never at all.





Spacelike infinities enforce minimalsurfaces→ apparent horizons→ BHs.

~S0;M0; Ja0~S1;M1; Ja1~S2;M2; Ja2


Excision technique: do not evolve values inside a pure outflow boundary

Singularity avoiding slicings: Choose coordinates such that the physicalsingularity is approached only at infinite coordinate time – or never at all.







“Punctures” become the workhorse initial data of BBH simulations.

The (1/r) singularity in the metric was factored out analytically not only forinitial data, but also evolutions.







“Punctures” become the workhorse initial data of BBH simulations.

The (1/r) singularity in the metric was factored out analytically not only forinitial data, but also evolutions.



2005: BBH Phase Transition to Gold Rush . . .




−1.2 −0.6 0 0.6 1.2x/M

−1.2

−0.6

0

0.6

1.2

y/M

0 10 20 30 40t/m

−0.0004

−0.0002

0

0.0002

0.0004

ψ4(M/16) −ψ4 (M/24)(ψ4(M/24) −ψ4 (M/36))*(81/16)

0 20 40 60−0.02

−0.01

0

0.01

0.02

ψ4(

Re

part

of (

l=2,

m=

2))

M/16M/24M/36

Figure: Results from Frans Pretorius April 2005 talk and July 2005 PRL, Novemberresults from NASA Goddard and UT Brownsville.

“Everything done differently” than everbody else:

Generalized second order harmonic evolution + constraint damping.

BH-excision regions can move through grid.

Full AMR + implicit evolution + high resolution + lots of dissipation.




−1.2 −0.6 0 0.6 1.2x/M

−1.2

−0.6

0

0.6

1.2

y/M

0 10 20 30 40t/m

−0.0004

−0.0002

0

0.0002

0.0004

ψ4(M/16) −ψ4 (M/24)(ψ4(M/24) −ψ4 (M/36))*(81/16)

0 20 40 60−0.02

−0.01

0

0.01

0.02

ψ4(

Re

part

of (

l=2,

m=

2))

M/16M/24M/36

Figure: Results from Frans Pretorius April 2005 talk and July 2005 PRL, Novemberresults from NASA Goddard and UT Brownsville.

UT Brownsville and NASA Goddard, Nov. 2005: Small modifications to standardBSSN codes do the trick:

Do not factor out puncture singularity during evolution!

Do not keep the puncture location fixed – drop artificial gauge-terms freezingthe shift at the puncture to zero!

Forget what you have been told about how BBH simulations should be done!



Schwarzschild for the moving puncture generation:Trumpets, not wormholes!

The stationary solutions for NR slicing equations develop cylindrical asymptotics,end at the throat → actual singularity is milder than what has been factored out.

[Hannam, SH, Pollney, Brugmann, O Murchadha, PRL 99, 241102 (2007)

[Hannam, SH, O Murchadha, et al, J. Phys. Conf. Series Series 66 012047 (2007)]

[Hannam, SH, Ohme, Brugmann, O Murchadha, PRD 78, 064020 (2008)]

“Trumpet data” – a new paradigm for high precision numerical evolutions:



Schwarzschild for the moving puncture generation:Trumpets, not wormholes!

The stationary solutions for NR slicing equations develop cylindrical asymptotics,end at the throat → actual singularity is milder than what has been factored out.

[Hannam, SH, Pollney, Brugmann, O Murchadha, PRL 99, 241102 (2007)

[Hannam, SH, O Murchadha, et al, J. Phys. Conf. Series Series 66 012047 (2007)]

[Hannam, SH, Ohme, Brugmann, O Murchadha, PRD 78, 064020 (2008)]

“Trumpet data” – a new paradigm for high precision numerical evolutions:i+R

i+L

I +I +

I − I −

i−Ri−L

i0L i0

R

R=

2M

R=

2M

R = 0 i+R

i+L

I +I +

I − I −

i−Ri−L

i0L i0

R

R=

2M R=

2M

R = 0



Asymmetric beaming of radiation causes recoil!

Astrophysics: recoil > 2000 km/s kicks remnant out of large galaxies.

Analytical estimates: 50 – 1000’s of km/s.

Full GR, no spins: Vmax = 175± 11 km/s for m1 ≈ 3m2.

[Gonzalez, Sperhake, Brugmann, Hannam, SH, PRL 98, 091101 (2007)]

Kick of ≈ 2500 km/s for spins in orbital plane, due to up-down asymmetry inleading quadrupole mode! Fuels astrophysical theory & observation!

[Gonzalez, Hannam, Sperhake, Brugmann, SH, PRL 98, 231101 (2007)

0.15 0.2 0.25η

0

50

100

150

200

250

300

v (

km

/s)

Baker, et alCampanelliDamour and GopakumarHerrmann, et alSopuerta, et alBlanchet, et al

-4

-2

0

2

4

x

-4-2

02

4

y

-0.15

-0.1

-0.05

0

z-4

-2

0

2

4

x -0.15

-0.1

-










0.15 0.2 0.25η

0

50

100

150

200

250

300

v (

km

/s)











0.15 0.2 0.25η

0

50

100

150

200

250

300

v (

km

/s)


-10

0

10

-10

0

10

-4-2

024


Applications Interlude: GW detection

General Relativity is becoming observational science!

International network of interferometersfinished first science run @ designsensitivity in 11/2007, enhanceddetectors start to operate in July.

Effort for first GW detectionreminiscent of large high energyexperiments.

Advanced, 3rd generation & spacedetectors to operate in the style ofastronomical observatories.

3 hours of Advanced LIGO ≈ 1 year ofinitial LIGO.

Era of GW astronomy is only about 2PhD theses away . . .


Applications Interlude: GW detection

General Relativity is becoming observational science!

International network of interferometersfinished first science run @ designsensitivity in 11/2007, enhanceddetectors start to operate in July.

Effort for first GW detectionreminiscent of large high energyexperiments.

Advanced, 3rd generation & spacedetectors to operate in the style ofastronomical observatories.

3 hours of Advanced LIGO ≈ 1 year ofinitial LIGO.

Era of GW astronomy is only about 2PhD theses away . . .


Applications Ultrarelativistic Collisions

Ultrarelativistic Collisions

High velocity collisions of BHs and particles – possible relevance for LHC.

Do colliding particles form BHs? What is the cross-section? GW-energy released?

Idealized calculation: collision of two infinitely boosted particles described byAichelburg-Sexl metric. TS forms in collision of two such “shock-waves”.

Collisions of solitons built from massive complex scalar field (Choptuik &Pretorius) suggest generic BH formation at boosts higher than v ≈ .94c .

Similar calculations for BH collisions show very large energy releases > 20%.

Higher dimensions: currently feasible if symmetries reduce the problem to 3+1.

Larger effective dimensions with increased computational power and possiblyimproved algorithms (compare neutrino transport problem for supernovae).

What exactly should we compute, what physical conclusions can we draw fromsimplified models?


Applications Ultrarelativistic Collisions

Ultrarelativistic Collisions

High velocity collisions of BHs and particles – possible relevance for LHC.

Do colliding particles form BHs? What is the cross-section? GW-energy released?

Idealized calculation: collision of two infinitely boosted particles described byAichelburg-Sexl metric. TS forms in collision of two such “shock-waves”.

Collisions of solitons built from massive complex scalar field (Choptuik &Pretorius) suggest generic BH formation at boosts higher than v ≈ .94c .

Similar calculations for BH collisions show very large energy releases > 20%.

Higher dimensions: currently feasible if symmetries reduce the problem to 3+1.

Larger effective dimensions with increased computational power and possiblyimproved algorithms (compare neutrino transport problem for supernovae).

What exactly should we compute, what physical conclusions can we draw fromsimplified models?


Getting started

Getting started in NR: Literature

Some textbooks now available:

Bona et al.

Alcubierre

Baumgarte & Shapiro

Living Reviews, http://relativity.livingreviews.org:

BBH: Articles citing F. Pretorius, gr-qc/0507014 (351).

Interface to GW-detection: NINJA project, http://www.ninja-project.org/

Happy to provide more literature!


Getting started

NR - getting started

Toy models: dynamics in 1+1 D (spherical symmetry!), 2D AH finder,“apples” tests.

Characteristic (null) code for critical collapse in spherical symmetry.

In 1+1 or 2+1 D, using Mathematica, Matlab, python, . . . can be a greattool to numerically solve PDEs.

Full 3+1 simulations: Einstein toolkit worth a look.

Kranc should be relatively easy to modify to generate code for frameworksother than Cactus.

Don’t use spectral evolution methods unless you know what you are doing.

Make friends!


Getting started








Make friends!


Getting started








Make friends!


Getting started








Make friends!


Getting started








Make friends!


Getting started








Make friends!


Getting started








Make friends!


Getting started

Conclusions

In NR we typically solve the Einstein equations as PDEs in the time domain.

The complicated structure of the constraints, diffeomorphism freedom, andthe lack of quasilocal energy/momentum create unique problems for NR.

First solution of the two-body problem in full GR took 4 decades.

Simulations of BHs, neutron stars, boson stars or supernovas are nowroutinely performed in 3+1 dimensions.

Accurate tracking of the orbital phase of binary black holes is hoped todramatically increase the scientific gain from GW simulations.

NR is a small field, but there is a lot of interest to expand to new problems.


Getting started

Conclusions








Getting started

Conclusions








Getting started

Conclusions








Getting started

Conclusions








Getting started

Conclusions








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