Summary: A class of second-order numerical schemes for the compressible Euler equations is described, and their \(L^ 1\) stability (i.e., \(\rho\geq 0\), \(T\geq 0\)) is proved. Following Van Leer’s approach [B. van Leer, J. Comput. Phys. 32, 101-136 (1979)] the solution (\(\rho,u,\sqrt T\) here) is represented as piecewise linear functions. The necessity of a slope limitation appears naturally in the derivation of the schemes, but it can be less strict than the slope reconstructions usually used. These schemes are written in terms of explicit flux splitting formula and are naturally multidimensional in space; the upwinding is obtained through a very generalized notion of characteristics: the kinetic one.