  Lagrangian smoothed particle hydrodynamics (SPH) is a well-established
approach to model fluids in astrophysical problems, thanks to its geometric
flexibility and ability to automatically adjust the spatial resolution to the
clumping of matter. However, a number of recent studies have emphasized
inaccuracies of SPH in the treatment of fluid instabilities. The origin of
these numerical problems can be traced back to spurious surface effects across
contact discontinuities, and to SPH's inherent prevention of mixing at the
particle level. We here investigate a new fluid particle model where the
density estimate is carried out with the help of an auxiliary mesh constructed
as the Voronoi tessellation of the simulation particles instead of an adaptive
smoothing kernel. This Voronoi-based approach improves the ability of the
scheme to represent sharp contact discontinuities. We show that this eliminates
spurious surface tension effects present in SPH and that play a role in
suppressing certain fluid instabilities. We find that the new `Voronoi Particle
Hydrodynamics' described here produces comparable results than SPH in shocks,
and better ones in turbulent regimes of pure hydrodynamical simulations. We
also discuss formulations of the artificial viscosity needed in this scheme and
how judiciously chosen correction forces can be derived in order to maintain a
high degree of particle order and hence a regular Voronoi mesh. This is
especially helpful in simulating self-gravitating fluids with existing gravity
solvers used for N-body simulations.
