Towards a fully well-balanced and entropy-stable scheme for the Euler equations with gravity: General equations of state

TL;DR

This paper develops a new entropy-stable, well-balanced finite volume scheme for the Euler equations with gravity, applicable to general equations of state, ensuring positivity and stability across various thermodynamic models.

Contribution

It extends existing entropy-stable, well-balanced schemes to general equations of state, providing a second-order, positivity-preserving method for Euler equations with gravity.

Findings

Scheme is entropy-stable and positivity-preserving for all thermodynamic variables.

Numerical tests confirm the scheme's performance across six different equations of state.

The method effectively maintains well-balanced properties for gravitationally influenced flows.

Abstract

The present work concerns the derivation of a fully well-balanced Godunov-type finite volume scheme for the Euler equations with a gravitational potential based on an approximate Riemann solver in a one-dimensional framework. It is an extension to general equations of states of the entropy-stable and fully well-balanced scheme for ideal gases recently forwarded in [Berthon et al., 2025]. A second-order extension preserving the properties of the first-order scheme is given. The scheme is provably entropy-stable and positivity-preserving for all thermodynamic variables. Numerical test cases illustrate the performance and entropy stability of the new scheme, using six different equations of state as examples, four analytic and two tabulated ones.