  The standard formulation of the smoothed particle hydrodynamics (SPH) assumes
that the local density distribution is differentiable. This assumption is used
to derive the spatial derivatives of other quantities. However, this assumption
breaks down at the contact discontinuity. At the contact discontinuity, the
density of the low-density side is overestimated while that of the high-density
side is underestimated. As a result, the pressure of the low (high) density
side is over (under) estimated. Thus, unphysical repulsive force appears at the
contact discontinuity, resulting in the effective surface tension. This tension
suppresses fluid instabilities. In this paper, we present a new formulation of
SPH, which does not require the differentiability of density. Instead of the
mass density, we adopt the internal energy density (pressure), and its
arbitrary function, which are smoothed quantities at the contact discontinuity,
as the volume element used for the kernel integration. We call this new
formulation density independent SPH (DISPH). It handles the contact
discontinuity without numerical problems. The results of standard tests such as
the shock tube, Kelvin-Helmholtz and Rayleigh-Taylor instabilities, point like
explosion, and blob tests are all very favorable to DISPH. We conclude that
DISPH solved most of known difficulties of the standard SPH, without
introducing additional numerical diffusion or breaking the exact force symmetry
or energy conservation. Our new SPH includes the formulation proposed by
Ritchie & Thomas (2001) as a special case. Our formulation can be extended to
handle a non-ideal gas easily.
