Recent results with Goddard Recent results with Goddard AMR codesAMR codes

Dae-Il (Dale) ChoiDae-Il (Dale) ChoiNASA/Goddard, USRANASA/Goddard, USRA

CollaboratorsCollaboratorsJ. Centrella, J. Baker, J. van Meter, D. Fiske, B. Imbiriba J. Centrella, J. Baker, J. van Meter, D. Fiske, B. Imbiriba

(NASA/Goddard)(NASA/Goddard)J. D. Brown and L. Lowe (NCSU)J. D. Brown and L. Lowe (NCSU)

Supported by NASA ATP02-0043-0056Supported by NASA ATP02-0043-0056

PSU NR Lunch, APR 29, 2004PSU NR Lunch, APR 29, 2004

OutlineOutline

Codes/FeaturesCodes/Features Summary of past and present worksSummary of past and present works

Brill wavesBrill waves Binary black holesBinary black holes

FutureFuture

CodesCodes

““HahndolHahndol” ” (= “One-Stone”= “Ein-stein” in Korean)(= “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 FMR, AMR and Parallel (Paramesh): scalability

good.good. Puncture BHs, Waves.Puncture BHs, Waves.

AMRMG_3DAMRMG_3D (NCSU)(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 Initial data generator: Brill wave, 2BH, single

distorted BH.distorted BH.

Key Features: Key Features: SimplicitySimplicity

Simple grid structure: Simple grid structure: AA hierarchy of hierarchy of the the logically cartesianlogically cartesian grid grid blockblocks s with with identicalidentical structurestructure. Considerably . Considerably simplified data structure is known at compile time.simplified data structure is known at compile time.

Simple tree structure: Grid blocks Simple tree structure: Grid blocks are managed by simple are managed by simple tree structure which tracks the spatial relationships between tree structure which tracks the spatial relationships between blocksblocks..

Mesh refinement isMesh refinement is ““block-basedblock-based” and uses” and uses ““bisectionbisection” ” method method (De Zeeuw (De Zeeuw && Powell, 1993) Powell, 1993) Block is a basic “unit” for mesh-refinement and domain Block is a basic “unit” for mesh-refinement and domain

decompositiondecomposition.. No clusterer needed.No clusterer needed.

Simple communication patterns: Simple communication patterns: Blocks are distributed Blocks are distributed amongst available processors in ways which maximize block amongst available processors in ways which maximize block locality and minimize inter-processor communications.locality and minimize inter-processor communications.

May be May be crucial for parallel implementationcrucial for parallel implementation

Mesh Refinement WorksMesh Refinement Works

Fixed Mesh Refinement (FMR) Fixed Mesh Refinement (FMR) Study of refinement interface conditions with linear waves Study of refinement interface conditions with linear waves

[JCP [JCP 193193, 398 (2004) (physics/0307036)], 398 (2004) (physics/0307036)] Key Ideas: Quadratic interpolation combined with “flux” Key Ideas: Quadratic interpolation combined with “flux”

matching guarantees 2matching guarantees 2ndnd order convergence and minimizes order convergence and minimizes interface noises.interface noises.

Single puncture BH: Single puncture BH: thoroughthorough convergence study with 8 convergence study with 8 levels of FMR [gr-qc/0403048].levels of FMR [gr-qc/0403048].

Key results: OB at ~100M, high resolution (h ~ 1/64) near Key results: OB at ~100M, high resolution (h ~ 1/64) near puncture.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 Weak GW simulations [PRD 6262, 084039 (2000)], 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.

Brill WavesBrill Waves

Initial Data: Time symmetric (axiInitial Data: Time symmetric (axi--symmetric) Brill wave symmetric) Brill wave solution.solution.

B.C.: Octant + Sommerfeld outgoing except .B.C.: Octant + Sommerfeld outgoing except . First order shock avoidance slicing First order shock avoidance slicing [M. Al[M. Alccubierre, CQG 20, 607 ubierre, CQG 20, 607

(2003)], (2003)], . . AMR interface conditions: 2AMR interface conditions: 2ndnd order interpolation followed by order interpolation followed by

“flux” matching “flux” matching matching function and first derivatives of matching function and first derivatives of function.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 Physics: find the critical parameter, A*, and study the critical phenomena (& later, extend to non-axisymmetry).phenomena (& later, extend to non-axisymmetry). Previous estimate of the critical parameter: 4.7 < A*< 5.0 Previous estimate of the critical parameter: 4.7 < A*< 5.0 [M. [M.

Alcubierre, Alcubierre, et al,et al, PRD 61, 041501 (2000), Use 128^3 grids ]. PRD 61, 041501 (2000), Use 128^3 grids ]. Hahndol: Zooming into critical regime: current estimation Hahndol: Zooming into critical regime: current estimation 4.80 4.80

< A* < 4.85.< A* < 4.85.

0

Brill Wave: Preliminary resultsBrill Wave: Preliminary results

Dispersal for A < Dispersal for A < 4.84.8

Lapse collapses for Lapse collapses for A > 4.85A > 4.85

4.8 < A* < 4.85: 4.8 < A* < 4.85: results are results are sensitive to various sensitive to various parameters such as parameters such as location of outer location of outer boundary and boundary and resolution.resolution.

Brill Wave: Preliminary ResultsBrill Wave: Preliminary Results A = 4.84 A = 4.84 64 x 64 x 64 base grid 64 x 64 x 64 base grid

(h~0.125)(h~0.125) 3 additional levels 3 additional levels finest finest

resolution = 0.015625 resolution = 0.015625 (effective resolution of 512 x (effective resolution of 512 x 512 x 512 unigrid)512 x 512 unigrid)

Snapshots for lapse (on Z=0 Snapshots for lapse (on Z=0 plane)plane)

Working on to find AH to Working on to find AH to confirm BH formation.confirm BH formation.

Caution: Inadequate Caution: Inadequate resolution may give resolution may give completely wrong outcome!completely wrong outcome! Run with only 2 additional Run with only 2 additional

levels results in dispersal levels results in dispersal (finest resolution = (finest resolution = 0.3125)0.3125)

Further study is under Further study is under way.way.

Binary Black Hole Simulation (Head-on Binary Black Hole Simulation (Head-on collision)collision)

Initial DataInitial Data (time = 0) (time = 0) Simple cases can be done by hand: two equal mass non-Simple cases can be done by hand: two equal mass non-

spinning black holes with zero initial velocity.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 RefinementMesh 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.

Binary Black Hole Simulations (Mesh Binary Black Hole Simulations (Mesh Structure)Structure)

Mesh Refinement allows one to put outer boundary as far as Mesh Refinement allows one to put outer boundary as far as possible.possible.

Efficient distribution of grid points: more near black holes.Efficient distribution of grid points: more near black holes.

BBH Head-on collisionBBH 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. Mesh refinement calculations. (OB at 120M)(OB at 120M) ggxxxx on Z=0 plane. on Z=0 plane. Gauge wave followed by physical wave.Gauge wave followed by physical wave.

BBH Head-on collisionBBH Head-on collision

Coordinate conditions [gCoordinate conditions [gtxtx, g, gtttt].]. 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.

FutureFuture

Attacking both “astrophysics” and “physics” Attacking both “astrophysics” and “physics” problems.problems. Astrophysics: orbiting black hole binaries, distorted Astrophysics: orbiting black hole binaries, distorted

black holes black holes gravitational wave astrophysics. gravitational wave astrophysics. Physics: Brill wave, etc.Physics: Brill wave, etc.

Analysis tools for mesh refinementAnalysis tools for mesh refinement Horizon findersHorizon finders Invariants, GW extractionInvariants, GW extraction

Focus on LISA source modeling: GW extraction Focus on LISA source modeling: GW extraction for black holes binaries for black holes binaries Data analysis. Data analysis.