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

A Density Independent Formulation of Smoothed Particle Hydrodynamics