TESTING THE LIMITATIONS OF A CHARACTERISTIC NUMERICAL RELATIVITY CODE

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TESTING THE LIMITATIONS OF A CHARACTERISTIC
NUMERICAL RELATIVITY CODE

• Kevin Lai
• Department of Mathematical Sciences
• University of South Africa
• Eleventh Marcel Grossmann Meeting on General
  Relativity
• Berlin, July 23-29, 2006
• Collaborators N. Bishop, B. Szilagyi, C.
  Reisswig, J. Thornburg
• laicw_at_astro.unisa.ac.za
• http//gauss1.unisa.ac.za/cwlai/talks/talks.html

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Overview

• MOTIVATION for a characteristic code
• An overview of the CHARACTERISTIC FORMALISM
• CURRENT STATUS of the 3D code
• A brief outline of Bishops LINEARIZED ANALYTIC
  SOLUTIONS
• Numerical RESULTS
• Summary and future work

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Motivation for a Characteristic Code

• The most popular approach to NR is based on the
  31 FORMALISM, where spacetime is foliated into
  SPACELIKE hypersurfaces
• STABILITY problem code crashes before it evolves
  to generate physically interesting results
• Usually they need some sort of ARTIFICIAL OUTER
  BOUNDARY CONDITIONS, it may be hard to justify
  the consistency of the BCs with the system under
  consideration, and the accuracy of the results is
  in doubt
• The calculation of gravitational waves can only
  be performed at a FINITE DISTANCE FROM THE
  SOURCE, the validity of which depends on how
  large the computational domain is. Also, the
  observational results for gravitational waves is
  effectively at infinite distance from the source

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Motivation for a Characteristic Code

• Another approach to NR is based on the
  CHARACTERISTIC FORMALISM, where spacetime is
  foliated according to the CHARACTERISTIC
  STRUCTURE
• It has been demonstrated that characteristic code
  is capable of LONG-TERM EVOLUTION
• The entire spacetime can be COMPACTIFIED into the
  computational domain, PHYSICALLY MOTIVATED
  BOUNDARY CONDITIONS can be incorporated
• It employs the characteristics of the system of
  equations, it is therefore in a sense a more
  NATURAL choice of coordinate. The gravitational
  waves are evaluated at null infinity (Bondi
  et.al.)

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Motivation for a Characteristic Code

• The system of equations is decomposed into a
  hierarchical system of PDEs, an almost explicit
  integration scheme can be used, it is therefore
  likely to be more EFFICIENT than a 31 code
• Some astrophysical systems, say, an ORBITING
  COMPACT STAR CAPTURED BY A BLACK HOLE, has not
  been solved by the traditional 31 approach. On
  the other hand, the characteristic approach has
  been used to give some promising results

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Overview of Characteristic Formalism

• Our spacetime is foliated into a family of
  outgoing null hypersurfaces labeled by u
• The angular direction is labeled by xA
• Surface area radial coordinate labeled by r
• The Bondi-Sachs metric can be written as

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Overview of Characteristic Formalism

• Some metric components are represented by
  spin-weighted fields
• where qA is a complex dyad
• Einsteins equations are decomposed into
  hypersurface equations R11, qA R1A, hAB RAB for
  b, U, and W, evolution equation qA qB RAB for J,
  and constraint equations R0a . Angular
  derivatives are represented by the eth operators
• Initial data are J on u0. Inner boundary data
  for J, b, U, and W which need to satisfy the
  constraints at rrmin

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Current Status of the 3D Code

• The current 3D code is derived from the PITT code
  developed at Pittsburgh by L. Lehner, N. Bishop,
  R. Gomez et.al. using the characteristic
  formalism
• The code is rewritten using the framework
  provided by Cactus (B. Szilagyi, N. Bishop, C.
  Reisswig, J. Thornburg, C. W. Lai), and is
  capable of running in parallel on multiple
  processors. It has been tested on a cluster of 8
  processors, with a maximum resolution of 1613,
  and the CPU time roughly satisfy
• Two different versions 2-patch stereographic
  angular coordinates and 6-patch angular
  coordinates

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Current Status of the 3D Code

• The code is 2nd order convergent
• The PITT code has been used to model a MASSIVE
  PARTICLE ORBITING NEAR A SCHWARZSCHILD BLACK HOLE
  for small particle mass compared to the mass of
  the black hole (PRD68,084015, 2003)

• Computed inspiral Dr-0.016 Quadrupole formula
  Dr-2.98x10-6 Perturbation method Dr-4.75x10-6
• Computed rate of energy loss dE/du 2.669x10-13
• Quadrupole formula dE/du 1.10x10-16
• Perturbation method dE/du 1.75x10-16

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Brief Outline of the Linearized Solutions

• Construct linearized solutions using the same
  coordinates and metric variables as in the code
  so direct comparison between numerical and
  analytic solutions can be made. Intermediate
  quantities can be compared should there be any
  needs to trace where the discrepancies came from
• Consider a matter distribution in the form of a
  thin, low density, spherical shell (
  ) which may vary over the angular direction
  around a Schwarzschild black hole ( )
  or empty space ( )
• Density and metric variables are regarded as
  small quantities
• where and
  Schwarzschild (vacuum) spacetime corresponds to
  . All
  2nd order quantities are ignored

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Brief Outline of the Linearized Solutions

• For simplicity we assume the quantities have
  harmonic time dependence and the angular
  dependence is simply the spin-weighted spherical
  harmonics which are the eigenfunctions of the
  eth operators
• where

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Brief Outline of the Linearized Solutions

• Einstein equations are reduced to a system of
  ODEs, which can be solved analytically for cases
  where at least one of n or M vanishes. The news
  is given by
• It turns out b0 is a constant throughout the
  vacuum region, and the solutions in general
  depend on 3 (complex) parameters b0, C1, C2

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Brief Outline of the Linearized Solutions

• The solution for l2 and M0 is
• The news is given by

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Brief Outline of the Linearized Solutions

• The solution for l3 and M0 is
• The news is given by

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Numerical Results

• The code is 2nd order convergence

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Numerical Results

• Noise developed in the interpatch region

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Numerical Results

• Ghost points of the blue patch lie on the grid
  lines of the other (red, green, yellow, pink)
  grids
• Interpatch interpolation is effectively 1D, 4th
  order approximation of angular derivatives
  relatively straightforward to implement

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Numerical Results

• Comparison for 2-patch and 6-patch ( angular
  cells are roughly the same in all cases)

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Numerical Results
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Numerical Results
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Numerical Results
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Numerical Results
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Numerical Results
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Numerical Results

• Dependence of convergent factor on l

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Numerical Results

• Assume Nq a x lb
• Spectrum of a polytrope of radius 15km in an
  orbit at r10M around a Schwarzschild black hole
  of 10 solar mass
• Roughly speaking, to resolve up to l40, Nq115

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Summary and Future Work

• We have a 2nd order convergent (including the
  news) parallel code for characteristic evolution
• There is noise developed in the interpatch region
  in the 2-patch stereographic code which could be
  reduced by using the 6-patch method
• A crude criteria for the minimum resolution
  required for the simulation of compact star
  orbiting a giant black hole is obtained
• Actual simulation will be done soon!