1. Recent results with Goddard AMR codes Dae-Il (Dale) Choi NASA/Goddard, USRA Collaborators J. Centrella, J. Baker, J. van Meter, D. Fiske, B. Imbiriba (NASA/Goddard) J. D. Brown and L. Lowe (NCSU) Supported by NASA ATP02-0043-0056 PSU NR Lunch, APR 29, 2004

2. Outline Codes/Features Codes/Features Summary of past and present works Summary of past and present works Brill waves Brill waves Binary black holes Binary black holes Future Future

3. Codes “Hahndol” (= “One-Stone”= “Ein-stein” in Korean) “Hahndol” (= “One-Stone”= “Ein-stein” in Korean) Vacuum Evolution code: 3+1 BSSN. Vacuum Evolution code: 3+1 BSSN. Free evolution (BSSN gauges imposed). Free evolution (BSSN gauges imposed). FMR, AMR and Parallel (Paramesh): scalability good. FMR, AMR and Parallel (Paramesh): scalability good. Puncture BHs, Waves. Puncture BHs, Waves. AMRMG_3D (NCSU) AMRMG_3D (NCSU) Elliptic solver: Multi-grid. Elliptic solver: Multi-grid. Parallel, AMR support based on Paramesh. Parallel, AMR support based on Paramesh. Initial data generator: Brill wave, 2BH, single distorted BH. Initial data generator: Brill wave, 2BH, single distorted BH.

4. Key Features: Simplicity Simple grid structure: A hierarchy of the logically cartesian grid blocks with identical structure. Considerably simplified data structure is known at compile time. Simple grid structure: A hierarchy of the logically cartesian grid blocks with identical structure. Considerably simplified data structure is known at compile time. Simple tree structure: Grid blocks are managed by simple tree structure which tracks the spatial relationships between blocks. Simple tree structure: Grid blocks are managed by simple tree structure which tracks the spatial relationships between blocks. Mesh refinement is “ block-based ” and uses “ bisection ” method (De Zeeuw & Powell, 1993) Mesh refinement is “ block-based ” and uses “ bisection ” method (De Zeeuw & Powell, 1993) Block is a basic “ unit ” for mesh-refinement and domain decomposition. Block is a basic “ unit ” for mesh-refinement and domain decomposition. No clusterer needed. No clusterer needed. Simple communication patterns: Blocks are distributed amongst available processors in ways which maximize block locality and minimize inter- processor communications. Simple communication patterns: Blocks are distributed amongst available processors in ways which maximize block locality and minimize inter- processor communications.  May be crucial for parallel implementation

5. Mesh Refinement Works Mesh Refinement Works Fixed Mesh Refinement (FMR) Fixed Mesh Refinement (FMR) Study of refinement interface conditions with linear waves [JCP 193, 398 (2004) (physics/0307036)] Study of refinement interface conditions with linear waves [JCP 193, 398 (2004) (physics/0307036)] Key Ideas: Quadratic interpolation combined with “flux” matching guarantees 2 nd order convergence and minimizes interface noises. Key Ideas: Quadratic interpolation combined with “flux” matching guarantees 2 nd order convergence and minimizes interface noises. Single puncture BH: thorough convergence study with 8 levels of FMR [gr-qc/0403048]. Single puncture BH: thorough convergence study with 8 levels of FMR [gr-qc/0403048]. Key results: OB at ~100M, high resolution (h ~ 1/64) near puncture. Key results: OB at ~100M, high resolution (h ~ 1/64) near puncture. Already very helpful in 2BH simulations [work in progress]. Already very helpful in 2BH simulations [work in progress]. Distorted BH [work in progress, D. Fiske]. Distorted BH [work in progress, D. Fiske]. Adaptive Mesh Refinement (AMR) Adaptive Mesh Refinement (AMR) Weak GW simulations [PRD 62, 084039 (2000)] Weak GW simulations [PRD 62, 084039 (2000)] 2-level; fine grid tracking the waves. 2-level; fine grid tracking the waves. Brill wave simulations [work in progress] Brill wave simulations [work in progress] Zooming into critical regime. Zooming into critical regime.

6. Brill Waves Initial Data: Time symmetric (axi-symmetric) Brill wave solution. Initial Data: Time symmetric (axi-symmetric) Brill wave solution. B.C.: Octant + Sommerfeld outgoing except. B.C.: Octant + Sommerfeld outgoing except. First order shock avoidance slicing [M. Alcubierre, CQG 20, 607 (2003)],. First order shock avoidance slicing [M. Alcubierre, CQG 20, 607 (2003)],. AMR interface conditions: 2 nd order interpolation followed by “flux” matching  matching function and first derivatives of function. AMR interface conditions: 2 nd order interpolation followed by “flux” matching  matching function and first derivatives of function. Adaptive regridding based on the first derivatives of variables. Adaptive regridding based on the first derivatives of variables. Physics: find the critical parameter, A*, and study the critical phenomena (& later, extend to non-axisymmetry). Physics: find the critical parameter, A*, and study the critical phenomena (& later, extend to non-axisymmetry). Previous estimate of the critical parameter: 4.7 < A*< 5.0 [M. Alcubierre, et al, PRD 61, 041501 (2000), Use 128^3 grids ]. Previous estimate of the critical parameter: 4.7 < A*< 5.0 [M. Alcubierre, et al, PRD 61, 041501 (2000), Use 128^3 grids ]. Hahndol: Zooming into critical regime: current estimation  4.80 < A* < 4.85. Hahndol: Zooming into critical regime: current estimation  4.80 < A* < 4.85.

7. Brill Wave: Preliminary results Dispersal for A < 4.8 Dispersal for A < 4.8 Lapse collapses for A > 4.85 Lapse collapses for A > 4.85 4.8 < A* < 4.85: results are sensitive to various parameters such as location of outer boundary and resolution. 4.8 < A* < 4.85: results are sensitive to various parameters such as location of outer boundary and resolution.

8. Brill Wave: Preliminary Results A = 4.84 A = 4.84 64 x 64 x 64 base grid (h~0.125) 64 x 64 x 64 base grid (h~0.125) 3 additional levels  finest resolution = 0.015625 (effective resolution of 512 x 512 x 512 unigrid) 3 additional levels  finest resolution = 0.015625 (effective resolution of 512 x 512 x 512 unigrid) Snapshots for lapse (on Z=0 plane) Snapshots for lapse (on Z=0 plane) Working on to find AH to confirm BH formation. Working on to find AH to confirm BH formation. Caution: Inadequate resolution may give completely wrong outcome! Caution: Inadequate resolution may give completely wrong outcome! Run with only 2 additional levels results in dispersal (finest resolution = 0.3125) Run with only 2 additional levels results in dispersal (finest resolution = 0.3125) Further study is under way. Further study is under way.

9. Binary Black Hole Simulation (Head-on collision) Initial Data (time = 0) Initial Data (time = 0) Simple cases can be done by hand: two equal mass non-spinning black holes with zero initial velocity. Simple cases can be done by hand: two equal mass non-spinning black holes with zero initial velocity. Spatial metric on 3d spacelike hypersurface, Spatial metric on 3d spacelike hypersurface, Evolution (time > 0) Evolution (time > 0) Lapse condition (1+log) Lapse condition (1+log) Shift condition (Hyperbolic driver) Shift condition (Hyperbolic driver) Mesh Refinement Mesh Refinement Source region: scale ~ M, put more grid points. Source region: scale ~ M, put more grid points. Wavezone: scale ~ (10--100)M, put less grid points. Wavezone: scale ~ (10--100)M, put less grid points. Boundary of computational domain: ~ a few hundred M. Boundary of computational domain: ~ a few hundred M.

10. Binary Black Hole Simulations (Mesh Structure) Mesh Refinement allows one to put outer boundary as far as possible. Efficient distribution of grid points: more near black holes.

11. BBH Head-on collision Initial separation = 5M, M=2, Two event horizons initially separated. Initial separation = 5M, M=2, Two event horizons initially separated. Mesh refinement calculations. (OB at 120M) Mesh refinement calculations. (OB at 120M) g xx on Z=0 plane. g xx on Z=0 plane. Gauge wave followed by physical wave. Gauge wave followed by physical wave.

12. BBH Head-on collision Coordinate conditions [g tx, g tt ]. Coordinate conditions [g tx, g tt ]. Two black hole merges into a single black hole. Two black hole merges into a single black hole. Gauge wave comes out first. Gauge wave comes out first. Assume profile of a single black hole after merger. Assume profile of a single black hole after merger.

13. Future Attacking both “astrophysics” and “physics” problems. Attacking both “astrophysics” and “physics” problems. Astrophysics: orbiting black hole binaries, distorted black holes  gravitational wave astrophysics. Astrophysics: orbiting black hole binaries, distorted black holes  gravitational wave astrophysics. Physics: Brill wave, etc. Physics: Brill wave, etc. Analysis tools for mesh refinement Analysis tools for mesh refinement Horizon finders Horizon finders Invariants, GW extraction Invariants, GW extraction Focus on LISA source modeling: GW extraction for black holes binaries  Data analysis. Focus on LISA source modeling: GW extraction for black holes binaries  Data analysis.