Numerical AstrophysicsNumerical Astrophysicsat FAUSTat FAUST

Work (to be) done in collaboration withWork (to be) done in collaboration with

Christopher BeetleChristopher Beetle

Steve BruennSteve Bruenn

Warner MillerWarner Miller

Wolfgang TichyWolfgang Tichy

Pedro MarronettiPedro MarronettiFlorida Atlantic UniversityFlorida Atlantic University

November 4, 2004November 4, 2004

The Roads AheadThe Roads Ahead

Main ObjectivesMain Objectives

• To develop stable numerical algorithms for Numerical RelativityTo develop stable numerical algorithms for Numerical Relativity• To integrate General Relativity in Multi-dimensional Supernova CodesTo integrate General Relativity in Multi-dimensional Supernova Codes

Physical ApplicationsPhysical Applications

• The integration of realistic microphysics into GR Hydrodynamical The integration of realistic microphysics into GR Hydrodynamical simulations, to produce more realistic simulations of Compact-Object simulations, to produce more realistic simulations of Compact-Object Binaries and Supernova ExplosionsBinaries and Supernova Explosions

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Stable Algorithms in NR: Our Emotional BaggageStable Algorithms in NR: Our Emotional Baggage

The codes crash way too quickly due to:The codes crash way too quickly due to:

Unstable formulationsUnstable formulations: : exponentially growing modes are excited by the wrong exponentially growing modes are excited by the wrong combination of evolution equations and numerical implementationscombination of evolution equations and numerical implementations

Awful boundary conditionsAwful boundary conditions: : exponentially growing modes are excited by exponentially growing modes are excited by boundary conditions that have little to do with the differential constraints satisfied in the boundary conditions that have little to do with the differential constraints satisfied in the bulk of the gridbulk of the grid

Unphysical Initial DataUnphysical Initial Data: : the initial data set does not represent accurately the the initial data set does not represent accurately the dynamics of the astrophysical system under studydynamics of the astrophysical system under study

SingularitiesSingularities: : there are no easy ways to deal with moving black holesthere are no easy ways to deal with moving black holes

HydrodynamicsHydrodynamics: : open problems with hydro in classical fluid dynamics, like open problems with hydro in classical fluid dynamics, like conservation of angular momentum and shock handlingconservation of angular momentum and shock handling

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ADMADM• Arnowit, Deser, and Misner ’62Arnowit, Deser, and Misner ’62

Conformally Traceless FormulationsConformally Traceless Formulations• BSSN (Shibata & Nakamura ’95, Baumgarte & Shapiro ’99), Laguna & Shoemaker (Laguna & Shoemaker ’02)BSSN (Shibata & Nakamura ’95, Baumgarte & Shapiro ’99), Laguna & Shoemaker (Laguna & Shoemaker ’02)• They are (usually) not fully hyperbolic, but are more stable than ADMThey are (usually) not fully hyperbolic, but are more stable than ADM

Constraint Enforcing FormulationsConstraint Enforcing Formulations• 1-D and 2-D (Choptuik et al. ’93, ’03), 3-D in spherical coordinates (Bonazzola et al. ’03), Re-solving constraints 1-D and 2-D (Choptuik et al. ’93, ’03), 3-D in spherical coordinates (Bonazzola et al. ’03), Re-solving constraints

(Anderson & Matzner ’03), Constraints as evolution eqs. (Gentle et al. ’04)(Anderson & Matzner ’03), Constraints as evolution eqs. (Gentle et al. ’04)

Stable Algorithms in NR: FormulationsStable Algorithms in NR: Formulations

Symmetric Hyperbolic FormulationsSymmetric Hyperbolic Formulations• Einstein-Christoffel (Anderson & York ’99), KST (Kidder et al. ’01), Bona-Masso (Bona et al. ’99, ’03), Tiglio (Tiglio Einstein-Christoffel (Anderson & York ’99), KST (Kidder et al. ’01), Bona-Masso (Bona et al. ’99, ’03), Tiglio (Tiglio

et al. ’03)et al. ’03)• Well posed, fully hyperbolic, straightforward recipes for BCsWell posed, fully hyperbolic, straightforward recipes for BCs

Characteristic FormulationsCharacteristic Formulations• Bondi & Sachs (Bondi ’62, Sachs ’62), Single Black Hole (Gómez et al. ’98), Black Hole – Neutron Star (Bishop et al. Bondi & Sachs (Bondi ’62, Sachs ’62), Single Black Hole (Gómez et al. ’98), Black Hole – Neutron Star (Bishop et al.

’99, ’03)’99, ’03)• Problems with caustics, may still prove useful with NS systemsProblems with caustics, may still prove useful with NS systems

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How many of these formulations have been tested in How many of these formulations have been tested in binary simulations?binary simulations?

ADMADM and and BSSNBSSN

Why is this test important?Why is this test important?

Because it has been shown (many, many times) that the jump from highly Because it has been shown (many, many times) that the jump from highly symmetric problems to full three-dimensional simulations is NOT trivialsymmetric problems to full three-dimensional simulations is NOT trivial

Usually, the source of complications in these jumps resides in the Usually, the source of complications in these jumps resides in the numerical implementation and not in the analytical formulationnumerical implementation and not in the analytical formulation

Stable Algorithms in NR: FormulationsStable Algorithms in NR: Formulations

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Why Binary Neutron Stars?Why Binary Neutron Stars?They already are astrophysically important systemsThey already are astrophysically important systems

They do not posses a high degree of symmetryThey do not posses a high degree of symmetry

They do not present singularitiesThey do not present singularities

They do not present shocks (not during the inspiral phase, at least)They do not present shocks (not during the inspiral phase, at least)

HoweverHowever They require non-trivial initial data setsThey require non-trivial initial data sets

They require hydrodynamical evolution algorithmsThey require hydrodynamical evolution algorithms

They require very long runsThey require very long runs

Fortunately, we have some solutionsFortunately, we have some solutions

Ell-SolverEll-Solver: Initial Data sets for BNSs: Initial Data sets for BNSs

GRHydGRHyd: Full GR-Hydro code for time evolution: Full GR-Hydro code for time evolution

Stable Algorithms in NR: The BNS LaboratoryStable Algorithms in NR: The BNS Laboratory

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Based on the Conformally Flat Thin-Sandwich Approach (Based on the Conformally Flat Thin-Sandwich Approach (CFACFA) (Wilson & Mathews ) (Wilson & Mathews ’88, ’95). This formulation for the ID sets guarantees the circularity of the orbits.’88, ’95). This formulation for the ID sets guarantees the circularity of the orbits.

It uses a Multigrid Elliptic Solver and is fully parallelized.It uses a Multigrid Elliptic Solver and is fully parallelized.

It has been used extensively in quasi-equilibrium sequences (Marronetti et al. ’98, It has been used extensively in quasi-equilibrium sequences (Marronetti et al. ’98, ’99, Marronetti & Shapiro ’03) and binary evolutions (Duez et al. ’03, Marronetti et al. ’99, Marronetti & Shapiro ’03) and binary evolutions (Duez et al. ’03, Marronetti et al. ’04).’04).

It will be used as a stepping stone in the GR-Supernova project. It will be used as a stepping stone in the GR-Supernova project.

The Codes: The Codes: Ell-SolverEll-Solver

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It has been built using Cactus, to guarantee:It has been built using Cactus, to guarantee:• Efficient parallelizationEfficient parallelization• PortabilityPortability• Easy access for future collaboratorsEasy access for future collaborators

The code design allows for easy interchange of gravitational fields and The code design allows for easy interchange of gravitational fields and hydrodynamical formulations: each formulation will be implemented in different hydrodynamical formulations: each formulation will be implemented in different modules or modules or thornsthorns..

The Codes: The Codes: GRHydGRHyd

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The plot shows the evolution of the total angular momentum of the system vs. The plot shows the evolution of the total angular momentum of the system vs. time using a BSSN thorn for the gravitational fields and a van Leer thorn for the time using a BSSN thorn for the gravitational fields and a van Leer thorn for the Hydro evolution. Hydro evolution.

These two thorns were developed following the algorithms and parameters These two thorns were developed following the algorithms and parameters described in Duez et al. ’03 and Marronetti et al. ’04, to serve as a code test and as described in Duez et al. ’03 and Marronetti et al. ’04, to serve as a code test and as a base benchmark for the test of new formulations.a base benchmark for the test of new formulations.

The Codes: The Codes: GRHydGRHyd

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The plot shows a comparison of the speedup as a function of the number of The plot shows a comparison of the speedup as a function of the number of processors for our Cactus based GRHyd and for a DAGH based code. A BNS processors for our Cactus based GRHyd and for a DAGH based code. A BNS simulation in a 128x56^2 grid was employed for this comparison. Runs executed on simulation in a 128x56^2 grid was employed for this comparison. Runs executed on the IBM p690 Regatta Cluster “copper” at NCSA. the IBM p690 Regatta Cluster “copper” at NCSA.

The Codes: The Codes: GRHydGRHyd

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BNS LaboratoryBNS Laboratory

• Rotating FrameRotating Frame

• Pi + Equatorial SymmetriesPi + Equatorial Symmetries

• Corotating & Irrotational BNSCorotating & Irrotational BNS

• Small grids at low resolutionSmall grids at low resolution

• Runs can be performed in Runs can be performed in single processor workstationssingle processor workstations

Stable Algorithms in NR: Testing…Stable Algorithms in NR: Testing…

Red: Grid size 256 x 1282 (Marronetti et al. ’04)

Green: Grid size 64 x 322 (half res.)

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Example RunExample Run Green grid size, corotating BNS, BSSN thorn, van Leer thorn.Green grid size, corotating BNS, BSSN thorn, van Leer thorn.

Stable Algorithms in NR: Testing…Stable Algorithms in NR: Testing…

Rest Mass density

HC Violation

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RadHyd and the Conformally Flat Approximation (Ell-Solver)RadHyd and the Conformally Flat Approximation (Ell-Solver)

GR Supernova SimulationsGR Supernova Simulations

Comparison between a Full GR (Shibata & Sekiguchi ’04) and a CFA (Dimmelmeier et al. ’03) Comparison between a Full GR (Shibata & Sekiguchi ’04) and a CFA (Dimmelmeier et al. ’03) simulation of core collapse. The plots are taken from Shibata & Sekiguchi PRD simulation of core collapse. The plots are taken from Shibata & Sekiguchi PRD 6969 (2004) (2004) 084024 and show gravitational wave amplitudes for three different models.084024 and show gravitational wave amplitudes for three different models.

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We propose the simulation of BNS during the inspiral phase as a testing ground for new numerical algorithms.We propose the simulation of BNS during the inspiral phase as a testing ground for new numerical algorithms.

Small grids at low resolution make possible the use single processor workstations: an orbital period can be evolved in a couple of hours.Small grids at low resolution make possible the use single processor workstations: an orbital period can be evolved in a couple of hours.

Ideal for testing:Ideal for testing:• New Evolution FormulationsNew Evolution Formulations• New Boundary ConditionsNew Boundary Conditions• New Gauge FieldsNew Gauge Fields

Future Expansion: Merger simulation as a testing ground for new hydrodynamical algorithmsFuture Expansion: Merger simulation as a testing ground for new hydrodynamical algorithms

Conclusions Conclusions

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