  Particle methods are a ubiquitous tool for solving the Vlasov-Poisson
equation in comoving coordinates, which is used to model the gravitational
evolution of dark matter in an expanding universe. However, these methods are
known to produce poor results on idealized test problems, particularly at late
times, after the particle trajectories have crossed. To investigate this, we
have performed a series of one- and two-dimensional "Zel'dovich Pancake"
calculations using the popular Particle-in-Cell (PIC) method. We find that PIC
can indeed converge on these problems provided the following modifications are
made. The first modification is to regularize the singular initial distribution
function by introducing a small but finite artificial velocity dispersion. This
process is analogous to artificial viscosity in compressible gas dynamics, and,
as with artificial viscosity, the amount of regularization can be tailored so
that its effect outside of a well-defined region - in this case, the
high-density caustics - is small. The second modification is the introduction
of a particle remapping procedure that periodically re-expresses the dark
matter distribution function using a new set of particles. We describe a
remapping algorithm that is third-order accurate and adaptive in phase space.
This procedure prevents the accumulation of numerical errors in integrating the
particle trajectories from growing large enough to significantly degrade the
solution. Once both of these changes are made, PIC converges at second order on
the Zel'dovich Pancake problem, even at late times, after many caustics have
formed. Furthermore, the resulting scheme does not suffer from the unphysical,
small-scale "clumping" phenomenon known to occur on the Pancake problem when
the perturbation wave vector is not aligned with one of the Cartesian
coordinate axes.
