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

TL;DR

The paper develops a hierarchy of Asymptotically Entropy-Conservative (AEC) fluxes for the compressible Euler equations that preserve kinetic-energy and pressure equilibrium while allowing controlled, asymptotic reduction of entropy-production errors through purely algebraic, low-cost flux formulations. By combining an arithmetic mean for density with a harmonic (and optionally geometric) mean for internal energy, the authors derive a first-order Arith-Harmonic flux that preserves pressure equilibrium and remains provably compatible with PEP at all expansion orders; higher-order terms reduce entropy error further. The framework unifies and extends existing KEP/EC approaches by providing a tunable, computationally efficient alternative to logarithmic-mean EC fluxes, and demonstrates robustness and accuracy through density-wave and Taylor-Green vortex tests. The results suggest substantial potential for high-performance solvers requiring reliable entropy control in compressible flows, with a clear path to higher-order extensions via established Ranocha-based techniques.

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.

Figures (2)

• Figure 1: Density wave simulation using different numerical fluxes. On the left, time evolution of entropy integral: black continuous lines with circles represent the $A\rho$-$He$ scheme; red continuous lines with triangles represent the $A\rho$-$Ae$ scheme; blue with plus signs identifies the $A\rho$-$Ap$ scheme; green with squares is used for the geometric mean flux $G\rho$-$Ge$. Dashed lines represent the AEC${}^{(1)}$ (black and circles) and KEEP${}^{(1)}$ (red and triangles) schemes. On the right, comparison of the density and pressure at time $t=30$; the exact solution is represented by a blue dotted line, the cyan line with asterisk markers is the solution obtained using the Ranocha flux in Eq. \ref{['eq:Ranocha_Flux_eint']}; the solution using the $A\rho$-$Ae$ scheme is not shown, as the simulation had already diverged. The mesh is discretized in 61 nodes and $\textrm{CFL} = 0.01$.
• Figure 2: Time evolution of entropy integral for the inviscid Taylor-Green vortex test using different numerical fluxes: black continuous lines with circles represent the $A\rho$-$He$ scheme; red continuous lines with triangles represent the $A\rho$-$Ae$ scheme; blue with plus signs identifies the $A\rho$-$Ap$ scheme; green with squares is used for the geometric mean flux $G\rho$-$Ge$. Dashed lines represent the AEC${}^{(1)}$ (black with circles) and KEEP${}^{(1)}$ (red with triangles) schemes. The mesh is discretized using $32\times32\times32$ nodes; fourth-order accurate fluxes are used on the left figure, sixth-order is employed for the figure on the right. In both cases $\textrm{CFL} = 0.1$.