Carles BonaCarles BonaTomas Ledvinka Tomas Ledvinka Carlos PalenzuelaCarlos Palenzuela

Miroslav ZacekMiroslav Zacek

MexicoMexico, December 2003, December 2003

Checking AwA tests with Z4Checking AwA tests with Z4 ((comenzando la revolucion rapida)comenzando la revolucion rapida)

The Z4 system The Z4 system Physical Review D67,Physical Review D67, 104005 (2003)104005 (2003)

10 Field equations10 Field equations

RR + + ZZ + + ZZ= 8= 8 (T (T – T/2 g– T/2 g ) )

14 dynamical fields g14 dynamical fields g , , ZZ

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

violations

Z4 evolution equationsZ4 evolution equations ((tt - - LL) K) Kij ij = - = - iiddjj + + [ [ (3)(3)RRijij + + iiZZjj + + jjZZii

- 2 K- 2 K22ijij + ( + (trtrK - K - 22) K) Kijij - S - Sijij + ½ ( + ½ (trtrS - S - ) ) ijij ] ]

   ((tt - - LL) Z) Zii = = [ [k k (K(Kkk

ii - - trtrK K kkii) ) - 2 K- 2 Kii

kk Z Zkk

++ i i - - ii// - S - Sii ] ]

((tt - - LL) ) = = /2 [/2 [(3)(3)R + (R + (trtrK - K - 22) ) trtrK - K - trtr(K(K22) ) + + 2 2 kkZZkk – 2 Z – 2 Zk k kk// - 2 - 2] ]

nnZZ = = Z Z00

3+1 covariance:3+1 covariance:

t’t’ = f(t)= f(t) x’ = g(x,t)x’ = g(x,t)

(3+1)-covariant generalization:(3+1)-covariant generalization:

((tt - - LL) ) lnln = - = - f f ((trtrK -K - mm ))

Generalized harmonic slicingsGeneralized harmonic slicings

Strongly hyperbolic iff f>0 (harmonic, 1+log,...)

First order version of Z4First order version of Z4gr-qc/0307067gr-qc/0307067

1rst order variables1rst order variables(( , , ijij , K , Kij ij , , ,, ZZkk , A , Akk , D , Dkijkij))

AAkk kk((lnln)) DDkijkij ½½ kk ijij

more constraints!more constraints!

  supplementary evolution equations supplementary evolution equations tt D Dkij kij + + k k [[ K Kij ij ] = 0] = 0

tt A Ak k + + k k [ [ f f ((trtrK - m K - m ) ] = 0) ] = 0

Robust stability testRobust stability test

Full 3D code with random small initial data Full 3D code with random small initial data (almost linear regime --> theorem) and (almost linear regime --> theorem) and periodic boundariesperiodic boundaries

Finite differencing: Method of linesFinite differencing: Method of lines– Standard 3rd order Runge-Kutta in timeStandard 3rd order Runge-Kutta in time– 1st order systems: standard centered 2nd order in 1st order systems: standard centered 2nd order in

spacespace– 2nd order systems: there is an ambiguity (3 point 2nd order systems: there is an ambiguity (3 point

stencil or 5 point stencil?)stencil or 5 point stencil?)

Strong vs Weak HyperbolicityStrong vs Weak Hyperbolicity

(dt=0.03*dx) slope of weak hyperbolic systems grows with the resolution

ICN resultsICN results

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

At the very end everything blows At the very end everything blows upup

T ~ 5 A^(-1/3) for ADMT ~ 4 A^(-1/2) for weakly Z4T ~ A^(-3/2) for strongly Z4---cosmological collapse?

Suggestions to clarify RobustSuggestions 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

Gauge wavesGauge waves Go to http://stat.uib.esGo to http://stat.uib.es We can check the linear and nonlinear We can check the linear and nonlinear

regime, the numerical method, study regime, the numerical method, study the numerical instability…the numerical instability…

Change A=0.1 to A=0.5Change A=0.1 to A=0.5 Study with one fixed formulation the Study with one fixed formulation the

different numerical methods (second or different numerical methods (second or fourth order in space, 3 and 5 points fourth order in space, 3 and 5 points scheme for second order systems, scheme for second order systems, dissipation,….)dissipation,….)

Collapsing Gowdy wavesCollapsing Gowdy waves Cosmological solution (vacuum) with Cosmological solution (vacuum) with

periodic boundariesperiodic boundariesdsds22 = t = t--1/2 1/2 eeQQ/2/2 (-dt (-dt22 + dz + dz22) + t (e) + t (ePP dxdx22 + e + e--PP dydy22) )

PP(t,z), Q(t,z) periodic in z(t,z), Q(t,z) periodic in z (pp wave)(pp wave)

Harmonic slicingHarmonic slicing

t = tt = t00 exp(- exp(-ττ//ττ00)) Testing the source termsTesting the source terms

Lapse collapse (Harmonic slicing)Lapse collapse (Harmonic slicing)

Oscillation & CollapseOscillation & Collapse

Things starts to be different at 2000 crossing times..then evolve up to 10.000

Z3 parameter space: nZ3 parameter space: n

Studying the sources of the formulation (adding energy, redefining variables,...)

ConclusionsConclusions 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 Use RK3 for the tests to avoid dissipation effect that can

mask the formulationsmask the formulations Remove/replace the linear waves; they do not give any Remove/replace the linear waves; they do not give any

new informationnew information Be careful with the stencil scheme (3-5) if you use second Be careful with the stencil scheme (3-5) if you use second

order systems!! (do you want to test the formulation or order systems!! (do you want to test the formulation or the numerical method?)the numerical method?)

Change the gauge waves amplitude (A=0.1 to A=0.5 to Change the gauge waves amplitude (A=0.1 to A=0.5 to study a strong non linear regime)study a strong non linear regime)

Evolve the Gowdy up to 10.000 crossing timesEvolve the Gowdy up to 10.000 crossing times

Boundary test suggestionsBoundary test suggestions Robust stability with boundaries: define exactly the Robust stability with boundaries: define exactly the

domain, face-edge-corners,..domain, face-edge-corners,.. 2D radial gauge wave (or gauge wave packet) with 2D radial gauge wave (or gauge wave packet) with

boundaries; exact solution not known, but a lot of boundaries; exact solution not known, but a lot of things to see (constraint violation, reflections,...)!things to see (constraint violation, reflections,...)!

Static solution (without excision or too large gradients) Static solution (without excision or too large gradients) with boundaries (ideas, suggestions?)with boundaries (ideas, suggestions?)

Wave moving in the static previous solution with Wave moving in the static previous solution with boundariesboundaries

General suggestionsGeneral suggestions We need more agressive (but isolated) tests with/without We need more agressive (but isolated) tests with/without

boundaries (it does not matter if we do not know the exact boundaries (it does not matter if we do not know the exact solution! Convergence tests are there)solution! Convergence tests are there)

We have to study in more detail some of the tests like gauge We have to study in more detail some of the tests like gauge waves to see what we can expectwaves to see what we can expect

Hurry, hurry, hurry! It is not difficult make all the tests, we can 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 not wait more than few months (2-3) to see the results, compare and take some results.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