8/18/2019 ESI_Husa_II http://slidepdf.com/reader/full/esihusaii 1/62 Introduction to Numerical Relativity II Numerics, Codes and Applications S. Husa University of the Balearic Islands, [email protected]August 3, 2010 S. Husa (UIB) Numerical Relativity Vienna 1 / 26
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WP: A unique solution exists at least for some time (when gauge is chosen),
depends continuously on initial data.
||u (t )|| ≤ Ke a 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 (e αt n ok, e αn not):
v n+1 = Q (t n, v n, v n+1)v n : ||v n|| ≤ Ke αt n ||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.”
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 stable
time 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!) . . .
∂ t β a − β a ,b β b = α2 (γ a − D a log α − haν Γ
ν ),
C a
≡ Γµ − x µ, C µ = 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 the
same way! Broad choice of variables!
Harmonic evolution appears prone to constraint violating modes, larger number of damping terms have been developed.
Harmonic lapse leads to evolutions where the spatial slice approaches a BH
singularity for large times.S. Husa (UIB) Numerical Relativity Vienna 10 / 26
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.
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.
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 of
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 of
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 of
PDE problems.Visualization tools.Alpaca tools for performance measuring and code testing.
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 of
PDE 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, . . .
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 of
PDE 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.
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 of
PDE 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.
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 of
PDE 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
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 of
PDE 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. Martin
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 the
space of initial dataUniversal critical solution is discretelyself-similar, corresponds to intermediateattractor in the language of dynamicalsystems. Mass scaling M BH ∝ |p − p |γ .
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).
Type II: analogous to second order phase transitions in statistical mechanics:
M BH ∝ |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 II
critical 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
Standard technique of math. relativityto treat black hole initial data: modelBH as nontrivial topology – add(compactified) infinities – “punctures”![Beig & O Murchadha, CQG (1994)]
Standard technique of math. relativityto treat black hole initial data: modelBH as nontrivial topology – add(compactified) infinities – “punctures”![Beig & O Murchadha, CQG (1994)]
Standard technique of math. relativityto treat black hole initial data: modelBH as nontrivial topology – add(compactified) infinities – “punctures”![Beig & O Murchadha, CQG (1994)]
Standard technique of math. relativityto treat black hole initial data: modelBH as nontrivial topology – add(compactified) infinities – “punctures”![Beig & O Murchadha, CQG (1994)]
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:
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
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).
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?