1. The waterbag method and Vlasov-Poisson equations in 1D: some examples S. Colombi (IAP, Paris) J. Touma (CAMS, Beirut)

2. Context Tradition: N-body - Poor resolution in phase-space -N–body relaxation Aims : direct resolution in phase-space. Now (almost ?) possible in with modern supercomputers Here: 1D gravity (2D phase-space)

3. Holes Suspect résonance x v Phase-space of a N-body simulation

4. Note : The waterbag method is very old Etc…

5. The waterbag method Exploits directly the fact that f[q(t),p(t),t]=constant along trajectories Suppose that f(q,p) independent of (q,p) in small patches (waterbags) (optimal configuration: waterbags are bounded by isocontours of f) It is needed to follow only the boundary of each patch, which can be sampled with an oriented polygon Polygons can be locally refined in order to give account of increasing complexity

6. Dynamics of sheets: 1D gravity Force calculation is reduced to a contour integral

7. Filamentation: need to add more and more points

8. Stationnary solution (Spitzer 1942) t=0

9. t=300

10. Ensemble of stationnary profiles

16. Relaxation of a Gaussian Few contours Many contours

21. Merger of 2 stationnary

22. Energy conservation

23. Pure waterbags: convergence study toward the cold case

24. Quasi stationary waterbag

30. Projected density: Singularity in r -2/3 Projected density: Singularity in r -1/2

31. The structure of the core

32. The logarithmic slope of the potential: Convergence study

33. Energy conservation Phase space volume conservation

34. Adiabatic invariant

36. Energies

37. Establishment of the central density profile: f=f 0 E -5/6 (Binney, 2004)

38. Effet of random perturbations

40. Energy conservation Phase space volume conservation

41. Effect of the perturbations on the slope

44. Refinement during runtime Normal case The curvature is changing sign TVD interpolation (no creation of artificial curvature terms) Note: in the small angle regime :

45. Time-step: standard Leapfrog (or predictor corrector if varying time step)

46. Better sampling of initial conditions: Isocontours Construction of the oriented polygon following isocontours of f using the marching cube algorithm Contour distribution computed such that the integral of (f sampled -f true ) 2 is bounded by a control parameter

47. Stationary solution (Spitzer 1942) Total mass Total energy