1. Carles Bona Tomas Ledvinka Carlos Palenzuela Miroslav Zacek Mexico, December 2003 Checking AwA tests with Z4 ( comenzando la revolucion rapida) ( comenzando la revolucion rapida)

2. 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

3. Z4 evolution equations (  t - L  ) K ij = -  i d j  +  [ (3) R ij +  i Z j +  j Z i (  t - L  ) K ij = -  i d j  +  [ (3) R ij +  i Z j +  j Z i - 2 K 2 ij + (trK - 2  ) K ij - S ij + ½ (trS -  )  ij ] - 2 K 2 ij + (trK - 2  ) K ij - S ij + ½ (trS -  )  ij ] (  t - L  ) Z i =  [  k (K k i - trK  k i ) - 2 K i k Z k (  t - L  ) Z i =  [  k (K k i - trK  k i ) - 2 K i k Z k +  i  -   i /  - S i ] (  t - L  )  =  /2 [ (3) R + (trK - 2  ) trK - tr(K 2 ) (  t - L  )  =  /2 [ (3) R + (trK - 2  ) trK - tr(K 2 ) + 2  k Z k – 2 Z k  k /  - 2  ]   n  Z  =  Z 0

4. 3+1 covariance: 3+1 covariance: t’ = f(t)x’ = g(x,t) (3+1)-covariant generalization: (3+1)-covariant generalization: (  t - L  ) ln  = -  f (trK - m  ) Generalized harmonic slicings Strongly hyperbolic iff f>0 (harmonic, 1+log,...)

5. First order version of Z4 gr-qc/0307067 1rst order variables 1rst order variables ( ,  ij, K ij, , Z k, A k, D kij ) A k   k (ln  )D kij  ½  k  ij more constraints! supplementary evolution equations supplementary evolution equations  t D kij +  k [  K ij ] = 0  t A k +  k [  f (trK - m  ) ] = 0

6. Robust stability test Full 3D code with random small initial data (almost linear regime --> theorem) and periodic boundaries Full 3D code with random small initial data (almost linear regime --> theorem) and periodic boundaries Finite differencing: Method of lines 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?)

7. Strong vs Weak Hyperbolicity (dt=0.03*dx) slope of weak hyperbolic systems grows with the resolution

8. ICN results (dt=0.03*dx) Numerical dissipation mask the linear growth: change the time integrator to RK3!!

9. 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?

10. 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

11. Gauge waves Go to http://stat.uib.es Go to http://stat.uib.es We can check the linear and nonlinear regime, the numerical method, study the numerical instability… We can check the linear and nonlinear regime, the numerical method, study the numerical instability… Change A=0.1 to A=0.5 Change A=0.1 to A=0.5 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,….) 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,….)

12. Collapsing Gowdy waves Cosmological solution (vacuum) with periodic boundaries Cosmological solution (vacuum) with periodic boundaries ds 2 = t - 1/2 e Q /2 (-dt 2 + dz 2 ) + t (e P dx 2 + e - P dy 2 ) P (t,z), Q(t,z) periodic in z(pp wave) Harmonic slicing Harmonic slicing t = t 0 exp(- τ / τ 0 ) Testing the source terms Testing the source terms

13. Lapse collapse (Harmonic slicing)

14. Oscillation & Collapse Things starts to be different at 2000 crossing times..then evolve up to 10.000

15. Z3 parameter space: n Studying the sources of the formulation (adding energy, redefining variables,...)

16. Conclusions Plot trK with robust and gauge waves should be enough Plot trK with robust and gauge waves should be enough Use RK3 for the tests to avoid dissipation effect that can mask the formulations 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 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?) 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 (A=0.1 to A=0.5 to study a strong non linear regime) Change the gauge waves amplitude (A=0.1 to A=0.5 to study a strong non linear regime) Evolve the Gowdy up to 10.000 crossing times Evolve the Gowdy up to 10.000 crossing times

17. Boundary test suggestions Robust stability with boundaries: define exactly the domain, face-edge-corners,.. 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,...)! 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?) Static solution (without excision or too large gradients) with boundaries (ideas, suggestions?) Wave moving in the static previous solution with boundaries Wave moving in the static previous solution with boundaries

18. 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 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 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. 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 systemSuggest new test If it is useful Everybody make the test and compare If it is not useful