  This study proposes a novel spatial discretization procedure for the
compressible Euler equations which guarantees entropy conservation at a
discrete level when an arbitrary equation of state is assumed. The proposed
method, based on a locally-conservative discretization, guarantees also the
spatial conservation of mass, momentum, and total energy and is kinetic
energy-preserving. In order to achieve the entropy-conservation property for an
arbitrary non-ideal gas, a general strategy is adopted based on the
manipulation of discrete balance equations through the imposition of global
entropy conservation and the use of a summation by parts rule. The procedure,
which is extended to an arbitrary order of accuracy, conducts to a general form
of the internal-energy numerical flux which results in a nonlinear function of
thermodynamic and dynamic variables and still admits the mass flux as a
residual degree of freedom. The effectiveness of the novel entropy-conservative
formulation is demonstrated through numerical tests making use of some of the
most popular cubic equations of state.
