Abstract 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 the “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 the “density.” We use this “density,” or pseudo-density, instead of the mass density, to formulate our SPH scheme. We call our new method SPH with smoothed pseudo-density, and we show that it is physically consistent and can handle discontinuities quite well.