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In this paper, we derive entropy estimates for a class of schemes for theEuler equations which present the following features: they are based on theinternal energy equation (eventually with a positive corrective term at therigh-hand-side so as to ensure consistency) and the possible upwinding isperformed with respect to the material velocity only. The implicit-in-timefirst-order upwind scheme satisfies a local entropy inequality. Ageneralization of the convection term is then introduced, which allows to limitthe scheme diffusion while ensuring a weaker property: the entropy inequalityis satisfied up to a remainder term which is shown to tend to zero with thespace and time steps, if the discrete solution is controlled in L $\infty$ andBV norms. The explicit upwind variant also satisfies such a weaker property, atthe price of an estimate for the velocity which could be derived from theintroduction of a new stabilization term in the momentum balance. Still for theexplicit scheme, with the above-mentioned generalization of the convectionoperator, the same result only holds if the ratio of the time to the space steptends to zero.

Entropy estimates for a class of schemes for the euler equations