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Recent results with Goddard AMR codes
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  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 • Summary of past and present works • Brill waves • Binary black holes • Future

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

  4. Key Features: Simplicity • Simple grid structure: A hierarchy of the logically cartesiangrid blocks with identicalstructure. 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. • Mesh refinement is“block-based” and uses“bisection”method (De Zeeuw & Powell, 1993) • Block is a basic “unit” for mesh-refinement and domain decomposition. • No clusterer needed. • 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 • Fixed Mesh Refinement (FMR) • Study of refinement interface conditions with linear waves [JCP 193, 398 (2004) (physics/0307036)] • Key Ideas: Quadratic interpolation combined with “flux” matching guarantees 2nd order convergence and minimizes interface noises. • Single puncture BH: thoroughconvergence study with 8 levels of FMR [gr-qc/0403048]. • Key results: OB at ~100M, high resolution (h ~ 1/64) near puncture. • Already very helpful in 2BH simulations [work in progress]. • Distorted BH [work in progress, D. Fiske]. • Adaptive Mesh Refinement (AMR) • Weak GW simulations [PRD 62, 084039 (2000)] • 2-level; fine grid tracking the waves. • Brill wave simulations [work in progress] • Zooming into critical regime.

  6. Brill Waves • Initial Data: Time symmetric (axi-symmetric) Brill wave solution. • B.C.: Octant + Sommerfeld outgoing except . • First order shock avoidance slicing [M. Alcubierre, CQG 20, 607 (2003)], . • AMR interface conditions: 2nd order interpolation followed by “flux” matching  matching function and first derivatives of function. • 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). • 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.

  7. Brill Wave: Preliminary results • Dispersal for A < 4.8 • 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.

  8. Brill Wave: Preliminary Results • A = 4.84 • 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) • Snapshots for lapse (on Z=0 plane) • Working on to find AH to confirm BH formation. • Caution: Inadequate resolution may give completely wrong outcome! • Run with only 2 additional levels results in dispersal (finest resolution = 0.3125) • Further study is under way.

  9. Binary Black Hole Simulation (Head-on collision) • Initial Data (time = 0) • Simple cases can be done by hand: two equal mass non-spinning black holes with zero initial velocity. • Spatial metric on 3d spacelike hypersurface, • Evolution (time > 0) • Lapse condition (1+log) • Shift condition (Hyperbolic driver) • Mesh Refinement • Source region: scale ~ M, put more grid points. • Wavezone: scale ~ (10--100)M, put less grid points. • 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. • Mesh refinement calculations. (OB at 120M) • gxx on Z=0 plane. • Gauge wave followed by physical wave.

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

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