  We analyze the conservation properties of various discretizations of the
system of compressible Euler equations for shock-free flows, with special focus
on the treatment of the energy equation and on the induced discrete equations
for other thermodynamic quantities. The analysis is conducted both
theoretically and numerically and considers two important factors
characterizing the various formulations, namely the choice of the energy
equation and the splitting used in the discretization of the convective terms.
The energy equations analyzed are total and internal energy, total enthalpy,
pressure, speed of sound and entropy. In all the cases examined the
discretization of the convective terms is made with locally conservative and
kinetic-energy preserving schemes. Some important relations between the various
formulations are highlighted and the performances of the various schemes are
assessed by considering two widely used test cases. Together with some popular
formulations from the literature, also new and potentially useful ones are
analyzed.
