  In this paper, we present a new formulation of smoothed particle
hydrodynamics (SPH), which, unlike the standard SPH (SSPH), is well-behaved at
the contact discontinuity. The SSPH scheme cannot handle discontinuities in
density (e.g. the contact discontinuity and the free surface), because it
requires that the density of fluid is positive and continuous everywhere. Thus
there is inconsistency in the formulation of the SSPH scheme at discontinuities
of the fluid density. To solve this problem, we introduce a new quantity
associated with particles and "density" of that quantity. This "density"
evolves through the usual continuity equation with an additional artificial
diffusion term, in order to guarantee the continuity of "density". We use this
"density" or pseudo density, instead of the mass density, to formulate our SPH
scheme. We call our new method as SPH with smoothed pseudo-density (SPSPH). We
show that our new scheme is physically consistent and can handle
discontinuities quite well.
