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Pennsylvania State University

Cauchy characteristic matching In cylindrical symmetry. Pennsylvania State University. Ulrich Sperhake. Joint work at Southampton University. Ray d’Inverno. James Vickers. Robert Sj ödin. Overview. ADM “3+1” formulation. Characteristic formulation. CCM in cylindrical symmetry.

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Pennsylvania State University

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  1. Cauchy characteristic matching In cylindrical symmetry Pennsylvania State University Ulrich Sperhake Joint work at Southampton University Ray d’Inverno James Vickers Robert Sjödin

  2. Overview • ADM “3+1” formulation • Characteristic formulation • CCM in cylindrical symmetry • 3+1 formulation • Characteristic formulation • Interface

  3. Ideal numerical code • fully non-linear 3D field + matter Eqs. • long term stability  • exact boundary conditions (infinity)  • extraction of grav. waves at infinity • proper treatment of singularities (excision, avoidance) • detailed description of matter (microphysics) • exact treatment of hydrodynamics (shock capturing) • high accuracy for signals with arbitrary amplitude

  4. ADM “3+1” formulation Arnowitt, Deser and Misner (1961) Foliate spacetime into 1-par. family of 3-dim. spacelike slices

  5. ”3+1” ADM formulation • Initial value problem • Field equations: 6 evolution Eqs. 3+1 constraints (conserved) • Dynamic variables: • Gauge variables:

  6. Advantages and drawbacks • “3+1” formulations preferred in regions of strong curvature • non-hyperbolicity of ADM  unclear stability properties => Modifications: introduce auxiliary variables => “BSSN”, hyperbolic formulations: appear to be more stable • Not clear how to compactify spacetime => 1) Interpretation of grav. waves at finite radii, 2) artificial boundary conditions at finite radii => spurious reflections, numerical noise

  7. Spurious reflections

  8. Characteristic formulation Bondi, Sachs (1962) • Foliate spacetime into 2-par. family of 2-dim. spacelike slices • One of the 2 families of curves threading the slices is null

  9. Characteristic formulation • Field equations: 2 evolution Eqs. 4 hypersurface Eqs. (in surfaces u=const) 3 supplementary Eqs., 1 trivial Eq. • compactification => 1) description of radiation at null infinity 2) Exact boundary conditions • Problem: Caustics in regions of strong curvature => Foliation breaks down “3+1” and char. formulation complement each other !

  10. Cauchy characteristic matching • “3+1” in interior region • char. In the outer region • interface at finite radius J. Winicour, Living Reviews, http://www.livingreviews.org

  11. How does it work in practice? • Cylindrically symmetric line element • Factor out z-Killing direction (Geroch decomposition) • Describe spacetime in terms of 2 scalar fields on 3-dim. quotient spacetime: Norm of the Killing vector Geroch potential

  12. Field equations □ □ □ Cauchy region: □ Characteristic region: , Compactification: => Null infinity at

  13. The interface

  14. Testing the code Xanthopoulos (1986)

  15. CCM versus ORC (Outgoing Radiation Condition) Cylindrical Gravitational Waves ORC (r=1) ORC (r=5) ORC (r=25) CCM

  16. Where to go from here? CCM in higher dimensions • axisymmetry (d’Inverno, Pollney) • 3 dim. (Bishop, Winicour et al.)

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