Summary: When the energy of a flow is largely kinetic, many conservative differencing schemes may fail by predicting non-physical states with negative density or internal energy. We describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the Godunov method is positively conservative. It is also shown that no scheme whose interface flux derives from a linearised Riemann solution can be positively conservative. Classes of data that will bring about the failure of such schemes are described. However, the Harten-Lax-van Leer scheme is positively conservative under certain conditions on the numerical wavespeeds, and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed.