  We revisit the integer lattice (IL) method to numerically solve the
Vlasov-Poisson equations, and show that a slight variant of the method is a
very easy, viable, and efficient numerical approach to study the dynamics of
self-gravitating, collisionless systems. The distribution function lives in a
discretized lattice phase-space, and each time-step in the simulation
corresponds to a simple permutation of the lattice sites. Hence, the method is
Lagrangian, conservative, and fully time-reversible. IL complements other
existing methods, such as N-body/particle mesh (computationally efficient, but
affected by Monte-Carlo sampling noise and two-body relaxation) and finite
volume (FV) direct integration schemes (expensive, accurate but diffusive). We
also present improvements to the FV scheme, using a moving mesh approach
inspired by IL, to reduce numerical diffusion and the time-step criterion.
Being a direct integration scheme like FV, IL is memory limited (memory
requirement for a full 3D problem scales as N^6, where N is the resolution per
linear phase-space dimension). However, we describe a new technique for
achieving N^4 scaling. The method offers promise for investigating the full 6D
phase-space of collisionless systems of stars and dark matter.
