1 / 16

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.

teagan
Download Presentation

Pennsylvania State University

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.)

More Related