Entropy conservative discretization of compressible Euler equations with an arbitrary equation of state

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.