Asymptotically entropy-conservative and kinetic-energy preserving numerical fluxes for compressible Euler equations

TL;DR

This paper introduces a hierarchy of computationally efficient numerical fluxes for the compressible Euler equations that preserve kinetic energy, pressure equilibrium, and are asymptotically entropy conservative, improving accuracy in entropy production.

Contribution

It develops a new class of fluxes based on algebraic means that are less costly and more accurate in entropy conservation than existing methods.

Findings

Numerical tests confirm entropy-preserving properties.

Harmonic mean fluxes outperform logarithmic mean in error reduction.

Geometric mean effectively reduces entropy evolution errors.

Abstract

This paper proposes a hierarchy of numerical fluxes for the compressible flow equations which are kinetic-energy and pressure equilibrium preserving and asymptotically entropy conservative, i.e., they are able to arbitrarily reduce the numerical error on entropy production due to the spatial discretization. The fluxes are based on the use of the harmonic mean for internal energy and only use algebraic operations, making them less computationally expensive than the entropy-conserving fluxes based on the logarithmic mean. The use of the geometric mean is also explored and identified to be well-suited to reduce errors on entropy evolution. Results of numerical tests confirmed the theoretical predictions and the entropy-conserving capabilities of a selection of schemes have been compared.