  Smoothed particle hydrodynamics is a particle-based, fully Lagrangian, method
for fluid-flow simulations. In this work, fundamental concepts of the method
are first briefly recalled. Then, we present a thorough comparison of three
different incompressibility treatments in SPH: the weakly compressible
approach, where a suitably-chosen equation of state is used; and two truly
incompressible methods, where the velocity field projection onto a
divergence-free space is performed. A noteworthy aspect of the study is that,
in each incompressibility treatment, the same boundary conditions are used (and
further developed) which allows a direct comparison to be made. Problems
associated with implementation are also discussed and an optimal choice of the
computational parameters has been proposed and verified. Numerical results show
that the present state-of-the-art truly incompressible method (based on a
velocity correction) suffer from density accumulation errors. To address this
issue, an algorithm, based on a correction for both particle velocities and
positions, is presented. The usefulness of this density correction is examined
and demonstrated in the last part of the paper.
