Numerical Astrophysics at FAUST

1. Numerical Astrophysicsat FAUST Pedro Marronetti Florida Atlantic University November 4, 2004 Work (to be) done in collaboration with Christopher Beetle Steve Bruenn Warner Miller Wolfgang Tichy

2. The Roads Ahead • Main Objectives • To develop stable numerical algorithms for Numerical Relativity • To integrate General Relativity in Multi-dimensional Supernova Codes • Physical Applications • The integration of realistic microphysics into GR Hydrodynamical simulations, to produce more realistic simulations of Compact-Object Binaries and Supernova Explosions 2

3. Stable Algorithms in NR: Our Emotional Baggage The codes crash way too quickly due to: • Unstable formulations: exponentially growing modes are excited by the wrong combination of evolution equations and numerical implementations • Awful boundary conditions: exponentially growing modes are excited by boundary conditions that have little to do with the differential constraints satisfied in the bulk of the grid • Unphysical Initial Data: the initial data set does not represent accurately the dynamics of the astrophysical system under study • Singularities: there are no easy ways to deal with moving black holes • Hydrodynamics: open problems with hydro in classical fluid dynamics, like conservation of angular momentum and shock handling 3

4. Stable Algorithms in NR: Formulations • ADM • Arnowit, Deser, and Misner ’62 • Conformally Traceless Formulations • BSSN (Shibata & Nakamura ’95, Baumgarte & Shapiro ’99), Laguna & Shoemaker (Laguna & Shoemaker ’02) • They are (usually) not fully hyperbolic, but are more stable than ADM • Constraint Enforcing Formulations • 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) • Symmetric Hyperbolic Formulations • Einstein-Christoffel (Anderson & York ’99), KST (Kidder et al. ’01), Bona-Masso (Bona et al. ’99, ’03), Tiglio (Tiglio et al. ’03) • Well posed, fully hyperbolic, straightforward recipes for BCs • Characteristic Formulations • Bondi & Sachs (Bondi ’62, Sachs ’62), Single Black Hole (Gómez et al. ’98), Black Hole – Neutron Star (Bishop et al. ’99, ’03) • Problems with caustics, may still prove useful with NS systems 4

5. Stable Algorithms in NR: Formulations How many of these formulations have been tested in binary simulations? ADM and BSSN Why is this test important? Because it has been shown (many, many times) that the jump from highly symmetric problems to full three-dimensional simulations is NOT trivial Usually, the source of complications in these jumps resides in the numerical implementation and not in the analytical formulation 5

6. Stable Algorithms in NR: The BNS Laboratory Why Binary Neutron Stars? They already are astrophysically important systems They do not posses a high degree of symmetry They do not present singularities They do not present shocks (not during the inspiral phase, at least) However They require non-trivial initial data sets They require hydrodynamical evolution algorithms They require very long runs Fortunately, we have some solutions Ell-Solver: Initial Data sets for BNSs GRHyd: Full GR-Hydro code for time evolution 6

7. The Codes: Ell-Solver • Based on the Conformally Flat Thin-Sandwich Approach (CFA) (Wilson & Mathews ’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 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. ’04). • It will be used as a stepping stone in the GR-Supernova project. 7

8. The Codes: GRHyd • It has been built using Cactus, to guarantee: • Efficient parallelization • Portability • Easy access for future collaborators • The code design allows for easy interchange of gravitational fields and hydrodynamical formulations: each formulation will be implemented in different modules or thorns. 8

9. The Codes: GRHyd • 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 Hydro evolution. • 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 a base benchmark for the test of new formulations. 9

10. The Codes: GRHyd • 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 simulation in a 128x56^2 grid was employed for this comparison. Runs executed on the IBM p690 Regatta Cluster “copper” at NCSA. 10

11. Stable Algorithms in NR: Testing… BNS Laboratory • Rotating Frame • Pi + Equatorial Symmetries • Corotating & Irrotational BNS • Small grids at low resolution • Runs can be performed in single processor workstations Red: Grid size 256 x 1282 (Marronetti et al. ’04) Green: Grid size 64 x 322 (half res.) 11

12. Stable Algorithms in NR: Testing… Example Run Green grid size, corotating BNS, BSSN thorn, van Leer thorn. HC Violation Rest Mass density 12

13. GR Supernova Simulations RadHyd and the Conformally Flat Approximation (Ell-Solver) 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 69 (2004) 084024 and show gravitational wave amplitudes for three different models. 13

14. Conclusions • 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. • Ideal for testing: • New Evolution Formulations • New Boundary Conditions • New Gauge Fields • Future Expansion: Merger simulation as a testing ground for new hydrodynamical algorithms 14