The waterbag method and Vlasov-Poisson equations in 1D:

some examples

S. Colombi (IAP, Paris)J. Touma (CAMS, Beirut)

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)

Holes

Suspect résonance

x

vPhase-space of a N-body simulation

Note : The waterbag method is very old

Etc…

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

Dynamics of sheets: 1D gravity

• Force calculation is reduced to a contour integral

Filamentation: need to add more and more points

Stationnary solution (Spitzer 1942)

t=0

t=300

Ensemble of stationnary profiles

Relaxation of a Gaussian

Few contours Many contours

Merger of 2 stationnary

Energy conservation

Pure waterbags: convergence study toward the cold case

Quasi stationarywaterbag

Projected density:Singularity in r-2/3

Projected density:Singularity in r-1/2

The structure of the core

The logarithmic slope of the potential:Convergence study

Energy conservation

Phase space volume conservation

Adiabatic invariant

Energies

Establishment of the central density profile: f=f0E-5/6 (Binney, 2004)

Effet of random perturbations

Energy conservation

Phase space volume conservation

Effect of the perturbations on the slope

Refinement during runtime

Normal case The curvature is changing sign

TVD interpolation (no creation of artificial curvature terms)

Note: in the small angle regime :

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

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 (fsampled-ftrue)2 is bounded by a control parameter

• Stationary solution (Spitzer 1942)

Total mass

Total energy