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

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Presentation transcript:

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 v Phase-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 stationary waterbag

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=f 0 E -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 (f sampled -f true ) 2 is bounded by a control parameter

Stationary solution (Spitzer 1942) Total mass Total energy