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