  We suggest a novel discretisation of the momentum equation for Smoothed
Particle Hydrodynamics (SPH) and show that it significantly improves the
accuracy of the obtained solutions. Our new formulation which we refer to as
relative pressure SPH, rpSPH, evaluates the pressure force in respect to the
local pressure. It respects Newtons first law of motion and applies forces to
particles only when there is a net force acting upon them. This is in contrast
to standard SPH which explicitly uses Newtons third law of motion continuously
applying equal but opposite forces between particles. rpSPH does not show the
unphysical particle noise, the clumping or banding instability, unphysical
surface tension, and unphysical scattering of different mass particles found
for standard SPH. At the same time it uses fewer computational operations. and
only changes a single line in existing SPH codes. We demonstrate its
performance on isobaric uniform density distributions, uniform density shearing
flows, the Kelvin-Helmholtz and Rayleigh-Taylor instabilities, the Sod shock
tube, the Sedov-Taylor blast wave and a cosmological integration of the Santa
Barbara galaxy cluster formation test. rpSPH is an improvement these cases. The
improvements come at the cost of giving up exact momentum conservation of the
scheme. Consequently one can also obtain unphysical solutions particularly at
low resolutions.
