  We consider explicit two-level three-point in space finite-difference schemes
for solving 1D barotropic gas dynamics equations. The schemes are based on
special quasi-gasdynamic and quasi-hydrodynamic regularizations of the system.
We linearize the schemes on a constant solution and derive the von Neumann type
necessary condition and a CFL type criterion (necessary and sufficient
condition) for weak conservativeness in $L^2$ for the corresponding
initial-value problem on the whole line. The criterion is essentially narrower
than the necessary condition and wider than a sufficient one obtained recently
in a particular case; moreover, it corresponds most well to numerical results
for the original gas dynamics system.
