Vlasov–Poisson: The waterbag method revisited

We revisit, with a view to refinement and generalization, the elegant waterbag method for the numerical treatment of Vlasov–Poisson equations. In this method, the phase space is decomposed into patches of constant density, and by exploiting Liouville’s theorem, the dynamics is reduced to the evolution of the boundary of these patches (waterbags). We follow the boundary using an adaptive, oriented polygon, and recover the force by circulating along this polygon. We discuss sampling of initial conditions with a set of oriented isocontours, and propose a new refinement procedure for accurate rendering of the stretching and folding polygon. Time evolution is naturally undertaken with symplectic algorithms. Tools, initially developed for systems of self-gravitating sheets, generalize naturally to spherically symmetric systems. We conclude with examples of both cases.