Finite-difference compatible entropy-conserving schemes for the compressible Euler equations

TL;DR

This paper develops a family of finite-difference schemes for the compressible Euler equations that conserve entropy, kinetic energy, and pressure equilibrium, validated through multiple test cases demonstrating their effectiveness.

Contribution

It introduces entropy-conserving finite-difference schemes that also preserve kinetic energy and pressure equilibrium, with broad applicability in finite-volume and finite-element methods.

Findings

Exact conservation of entropy demonstrated in tests.

Schemes preserve primary quantities and pressure equilibrium.

Validated on various 1D, 2D, and 3D flow problems.

Abstract

This paper introduces a family of entropy-conserving finite-difference discretizations for the compressible flow equations. In addition to conserving the primary quantities of mass, momentum, and total energy, the methods also preserve kinetic energy and pressure equilibrium. The schemes are based on finite-difference (FD) representations of the logarithmic mean, establishing and leveraging a broader link between linear and nonlinear two-point averages and FD forms. The schemes are locally conservative due to the summation-by-parts property and therefore admit a local flux form, making them applicable also in finite-volume and finite-element settings. The effectiveness of these schemes is validated through various test cases (1D Sod shock tube, 1D density wave, 2D isentropic vortex, 3D Taylor-Green vortex) that demonstrate exact conservation of entropy along with conservation of the primary quantities and preservation of pressure equilibrium.