  The kinetic theory of rarefied gases and numerical schemes based on the
Boltzmann equation have evolved to the cornerstone of non-equilibrium gas
dynamics. However, their counterparts in the dense regime remain rather exotic
for practical non-continuum scenarios. This problem is partly due to the fact
that long-range interactions arising from the attractive tail of molecular
potentials, lead to a computationally demanding Vlasov integral. This study
focuses on numerical remedies for efficient stochastic particle simulations
based on the Enskog-Vlasov kinetic equation. In particular, we devise a
Poisson-type elliptic equation that governs the underlying long-range
interactions. The idea comes through fitting a Green function to the molecular
potential, and hence deriving an elliptic equation for the associated
fundamental solution. Through this transformation of the Vlasov integral,
efficient Poisson-type solvers can be readily employed in order to compute the
mean field forces. Besides the technical aspects of different numerical schemes
for the treatment of the Vlasov integral, simulation results for the
evaporation of a liquid slab into the vacuum are presented. It is shown that
the proposed formulation leads to accurate predictions with a reasonable
computational cost.
