Vlasov-Poisson in 1D: waterbags

Abstract

We revisit in one dimension the waterbag method to solve numerically Vlasov-Poisson equations. In this approach, the phase-space distribution function f (x, v) is initially sampled by an ensemble of patches, the waterbags, where f is assumed to be constant. As a consequence of Liouville theorem it is only needed to follow the evolution of the border of these waterbags, which can be done by employing an orientated, self-adaptive polygon tracing isocontours of f . This method, which is entropy conserving in essence, is very accurate and can trace very well non linear instabilities as illustrated by specific examples.As an application of the method, we generate an ensemble of single waterbag simulations with decreasing thickness, to perform a convergence study to the cold case. Our measurements show that the system relaxes to a steady state where the gravitational potential profile is a power-law of slowly varying index β, with β close to 3/2 as found in the literature. However, detailed analysis of the properties of the gravitational potential shows that at the center, β > 1.54. Moreover, our measurements are consistent with the value β = 8/5 = 1.6 that can be analytically derived by assuming that the average of the phase-space density per energy level obtained at crossing times is conserved during the mixing phase. These results are incompatible with the logarithmic slope of the projected density profile β − 2 ≃ −0.47 obtained recently by Schulz et al. (2013) using a N -body technique. This sheds again strong doubts on the capability of N -body techniques to converge to the correct steady state expected in the continuous limit.

Vlasov–Poisson in 1D for initially cold systems: post-collapse Lagrangian perturbation theory

“…After this, the energetic properties of the approximate system deviate significantly from the simulation and probably from the true solution. Indeed, the close to stationary behavior of function fE(E) at late times -modulo the small scales fluctuations due to the cold nature of the system-seems to be a robust numerical result according to the convergence analysis performed in Colombi & Touma (2014). In particular, the deviation between theory and measurements observed on right panel of Fig.…”

“…Indeed, it has been shown in Colombi & Touma (2014) (hereafter, CT) that this function becomes nearly stationary for n > ∼ 3 when studied as a function of E − Emin, where Emin is the (time dependent) minimum of energy coinciding with the minimum of the potential. Furthermore, it was found by CT that the logarithmic slope of fE(E) was consistent, even at late times, with the one predicted at crossing times for small values of E − Emin.…”

“…The method employed to compute fE(E) from equation (104) is explained in details in Colombi & Touma (2014). 2500 linear bins were used in the interval [Emin, Emax], where Emin and Emax are respectively the minimum and the maximum of energy inside the waterbag.…”

“…To do this we focus on the simple case of one dimensional gravity and try to compute analytically the evolution of a single, initially cold and smooth structure, following the footsteps of many previous works in the literature (see, e.g., Doroshkevich et al 1980;Melott 1983;Shandarin & Zeldovich 1989;Rouet, Feix, & Navet 1990;Gurevich & Zybin 1995;Binney 2004;Schulz et al 2013;Colombi & Touma 2014).…”

A ‘metric’ semi-Lagrangian Vlasov–Poisson solver

“…To demonstrate the potential improvements brought by our code Vlamet compared to traditional semi-Lagrangian approaches, we compare our algorithm to the traditional splitting method of Cheng and Knorr (1976) with third order spline interpolation. Results are also tested against simulations performed with the entropy conserving waterbag code of Colombi & Touma (2014). The waterbag scheme, extremely accurate but very costly, is meant to provide a supposedly "exact" solution.…”