Logarithmic mean approximation in improving entropy conservation in KEEP scheme with pressure equilibrium preservation property for compressible flows

This study develops non-dissipative and robust spatial discretizations in kinetic-energy and entropy preserving (KEEP) schemes by improving the entropy conservation property, while maintaining the pressure-equilibrium-preservation (PEP) property. A main focus of this study is the approximation of the logarithmic mean, involved in the entropy-conservative numerical fluxes in the mass and energy equations. To seek suitable approximations of the logarithmic mean with the KEEP and PEP properties, we first derive the PEP condition for general entropy-conservative numerical fluxes. Then, we evaluate the entropy conservation errors for different approximations of the logarithmic mean. The present theoretical analyses reveal that the use of the geometric mean improves the entropy conservation error better than the other means. Given this theoretical result, we derive an asymptotic expansion of the logarithmic mean based on the geometric mean, which yields a smaller entropy conservation error than the existing expansions based on the arithmetic mean at each truncation order. Numerical experiments for one-dimensional density wave advection, two-dimensional isentropic vortex, three-dimensional compressible inviscid Taylor–Green vortex, and stationary normal shock demonstrate the validity of the present theoretical analyses.