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The waterbag method and Vlasov-Poisson equations in 1D:
some examples
S. Colombi (IAP, Paris)J. Touma (CAMS, Beirut)
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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)
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Holes
Suspect résonance
x
vPhase-space of a N-body simulation
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Note : The waterbag method is very old
Etc…
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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
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Dynamics of sheets: 1D gravity
• Force calculation is reduced to a contour integral
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Filamentation: need to add more and more points
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Stationnary solution (Spitzer 1942)
t=0
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t=300
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Ensemble of stationnary profiles
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Relaxation of a Gaussian
Few contours Many contours
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Merger of 2 stationnary
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Energy conservation
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Pure waterbags: convergence study toward the cold case
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Quasi stationarywaterbag
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Projected density:Singularity in r-2/3
Projected density:Singularity in r-1/2
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The structure of the core
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The logarithmic slope of the potential:Convergence study
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Energy conservation
Phase space volume conservation
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Adiabatic invariant
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Energies
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Establishment of the central density profile: f=f0E-5/6 (Binney, 2004)
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Effet of random perturbations
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Energy conservation
Phase space volume conservation
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Effect of the perturbations on the slope
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Refinement during runtime
Normal case The curvature is changing sign
TVD interpolation (no creation of artificial curvature terms)
Note: in the small angle regime :
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Time-step: standard Leapfrog(or predictor corrector if varying time step)
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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
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• Stationary solution (Spitzer 1942)
Total mass
Total energy