Carles%20Bona

1
Checking AwA tests with Z4 (comenzando la
revolucion rapida)

• Carles Bona
• Tomas Ledvinka
• Carlos Palenzuela
• Miroslav Zacek
• Mexico, December 2003

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The Z4 system Physical Review D67, 104005 (2003)

• 10 Field equations
• R?? ??Z? ??Z? 8? (T?? T/2 g?? )
• 14 dynamical fields g?? , Z?

Covariant formulation with Z quantities to
monitorize (and maybe enforce in the future) the
constraint violations
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Z4 evolution equations

• (?t - L?) Kij - ?idj ? ? (3)Rij ?iZj
  ?jZi
• - 2 K2ij (trK - 2?) Kij - Sij ½ (trS -
  ?) ?ij
•  
• (?t - L?) Zi ? ?k (Kki - trK ?ki) - 2 Kik Zk
• ?i ? - ? ?i/? - Si
• (?t - L?) ? ?/2 (3)R (trK - 2?) trK -
  tr(K2)
• 2 ?kZk 2 Zk ?k/? - 2?

? ? n?Z? ? Z0
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Generalized harmonic slicings

• 31 covariance
• t f(t) x g(x,t)
• (31)-covariant generalization
• (?t - L?) ln ? - ? f (trK - m ? )

Strongly hyperbolic iff fgt0 (harmonic, 1log,...)
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First order version of Z4gr-qc/0307067

• 1rst order variables
• (? , ?ij , Kij , ? , Zk , Ak , Dkij)
• Ak ? ?k(ln?) Dkij ? ½ ?k ?ij
• more constraints!
•  supplementary evolution equations
• ?t Dkij ?k ? Kij 0
• ?t Ak ?k ? f (trK - m ?) 0

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Robust stability test

• Full 3D code with random small initial data
  (almost linear regime --gt theorem) and periodic
  boundaries
• Finite differencing Method of lines
• Standard 3rd order Runge-Kutta in time
• 1st order systems standard centered 2nd order in
  space
• 2nd order systems there is an ambiguity (3 point
  stencil or 5 point stencil?)

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Strong vs Weak Hyperbolicity
(dt0.03dx) slope of weak hyperbolic systems
grows with the resolution
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ICN results
(dt0.03dx) Numerical dissipation mask the
linear growth change the time integrator to RK3!!
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At the very end everything blows up
T 5 A(-1/3) for ADM T 4 A(-1/2) for
weakly Z4 T A(-3/2) for strongly
Z4---cosmological collapse?
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Suggestions to clarify Robust

• Changing the time integrator to RK3 and/or using
  smaller courant factor
• Using appropiate initial data (distribute
  energy) for clear convergence tests
• 2nd order systems using the 5 points scheme in
  order to recover the theorem results or at least
  comparing with the known results with 3 points
  scheme
• Plotting trK is enough to see if it works or not
  

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Gauge waves

• Go to http//stat.uib.es
• We can check the linear and nonlinear regime, the
  numerical method, study the numerical
  instability
• Change A0.1 to A0.5
• Study with one fixed formulation the different
  numerical methods (second or fourth order in
  space, 3 and 5 points scheme for second order
  systems, dissipation,.)

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Collapsing Gowdy waves

• Cosmological solution (vacuum) with periodic
  boundaries
• ds2 t-1/2 eQ/2 (-dt2 dz2) t (eP dx2 e-P
  dy2)
• P(t,z), Q(t,z) periodic in z (pp wave)
• Harmonic slicing
• t t0 exp(-t/t0)
• Testing the source terms

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Lapse collapse (Harmonic slicing)
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Oscillation Collapse
Things starts to be different at 2000 crossing
times..then evolve up to 10.000
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Z3 parameter space n
Studying the sources of the formulation (adding
energy, redefining variables,...)
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Conclusions

• Plot trK with robust and gauge waves should be
  enough
• Use RK3 for the tests to avoid dissipation effect
  that can mask the formulations
• Remove/replace the linear waves they do not give
  any new information
• Be careful with the stencil scheme (3-5) if you
  use second order systems!! (do you want to test
  the formulation or the numerical method?)
• Change the gauge waves amplitude (A0.1 to A0.5
  to study a strong non linear regime)
• Evolve the Gowdy up to 10.000 crossing times

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Boundary test suggestions

• Robust stability with boundaries define exactly
  the domain, face-edge-corners,..
• 2D radial gauge wave (or gauge wave packet) with
  boundaries exact solution not known, but a lot
  of things to see (constraint violation,
  reflections,...)!
• Static solution (without excision or too large
  gradients) with boundaries (ideas, suggestions?)
• Wave moving in the static previous solution with
  boundaries

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General suggestions

• We need more agressive (but isolated) tests
  with/without boundaries (it does not matter if we
  do not know the exact solution! Convergence tests
  are there)
• We have to study in more detail some of the tests
  like gauge waves to see what we can expect
• Hurry, hurry, hurry! It is not difficult make all
  the tests, we can not wait more than few months
  (2-3) to see the results, compare and take some
  results.

Check with hyperbolic system
Suggest new test
If it is not useful
If it is useful
Everybody make the test and compare