  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 $\beta$, with $\beta$ close to $3/2$ as found in the literature. However,
detailed analysis of the properties of the gravitational potential shows that
at the center, $\beta > 1.54$. Moreover, our measurements are consistent with
the value $\beta=8/5=1.6$ 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 $\beta-2 \simeq -0.47$
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.
