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

TL;DR

This work advances numerical simulation of the one-dimensional Euler equations with gravity by constructing a fully well-balanced, entropy-stable Godunov-type finite volume scheme that supports general equations of state. The core method is an approximate Riemann solver with two intermediate states, augmented by a gravity-consistent discretization of the source term to preserve moving and hydrostatic equilibria while guaranteeing positivity of thermodynamic variables and a discrete entropy inequality. The authors derive detailed expressions for the intermediate states, density corrections, and source-term approximations, and provide a second-order extension using equilibrium-variable reconstruction and a two-stage time integrator. Numerical tests across four cubic EOS and two tabulated EOS (CoolProp) demonstrate accurate resolution of moving equilibria and perturbations, robust entropy decay, and superior performance compared with a non-well-balanced HLL solver, confirming the method’s practicality for realistic EOS in atmospheric and astrophysical contexts.

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.

Figures (17)

• Figure 1: Left panel: One possible wave configuration, with $u > 0$, of the Euler equations with gravity \ref{['eq:EulerG']}. Right panel: Wave structure of the approximate Riemann solver.
• Figure 2: Drawings of the functions $\psi_1$ (top left panel), $\mathcal{M}$ (top right panel), and $\psi$ (bottom panel).
• Figure 3: Experimental order of convergence and $L^2$ errors between the exact solution \ref{['eq:exact_sol']} and its approximation by the HLL, FWB1 and FWB2 schemes, for each of the six EOSs under consideration.
• Figure 4: Efficiency curves (error w.r.t. CPU time) for the three schemes, on the exact solution from \ref{['sec:accuracy']}, and for each of the six EOSs under consideration. Meshes with $({2^\ell})_{\ell \in \{4, \dots, 10\}}$ cells are used for the HLL and FWB1 schemes, and $({2^\ell})_{\ell \in \{4, \dots, 8\}}$ cells are considered for the FWB2 scheme (more cells would yield very low errors, making it hard to plot the difference between the other two schemes).
• Figure 5: Perturbation of equilibrium state, described in \ref{['sec:perturbation_bump']}: Relative density difference $\eta_\rho = (\rho_\text{eq}(x) - \rho)/\rho_\text{eq}(x)$ with respect to the equilibrium density $\rho_\text{eq}(x)$, where $\rho$ is obtained by the FWB1 scheme with a mesh made of $50$ and 7500.0 discretization cells and FWB2 scheme on $50$ cells, using the six EOSs under consideration.

Theorems & Definitions (20)

• Definition 3.1
• Lemma 3.2: Sufficient condition for well-balancedness
• Definition 3.3: Entropy stability
• Definition 3.4: Positivity preservation
• Lemma 3.5: Density positivity
• Lemma 3.6: Positivity of internal energy
• proof
• Lemma 3.7
• proof
• Lemma 3.8