Towards a fully well-balanced and entropy-stable scheme for the Euler equations with gravity: preserving isentropic steady solutions

TL;DR

The work addresses the robust numerical treatment of the Euler equations with gravity, focusing on positivity, entropy stability, and exact preservation of moving and hydrostatic equilibria. It introduces a fully well-balanced, entropy-stable Godunov-type scheme based on a two-intermediate-state approximate Riemann solver that reduces to the standard HLL solver in the gravity-free limit. The method preserves density and pressure positivity, satisfies discrete entropy inequalities, and exactly captures moving equilibria and isentropic hydrostatic states, with a higher-order extension and a 2D Cartesian extension. Numerical tests in 1D and 2D validate accuracy, well-balancedness, and the scheme's ability to resolve small perturbations around equilibria and Riemann problems under gravity, highlighting practical robustness for atmospheric and astrophysical flows.

Abstract

The present work concerns the derivation of a numerical scheme to approximate weak solutions of the Euler equations with a gravitational source term. The designed scheme is proved to be fully well-balanced since it is able to exactly preserve all moving equilibrium solutions, as well as the corresponding steady solutions at rest obtained when the velocity vanishes. Moreover, the proposed scheme is entropy-preserving since it satisfies all fully discrete entropy inequalities. In addition, in order to satisfy the required admissibility of the approximate solutions, the positivity of both approximate density and pressure is established. Several numerical experiments attest the relevance of the developed numerical method. An extension to two-dimensional problems is given, applying the one-dimensional framework direction by direction on Cartesian grids.

Figures (13)

• 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: Juxtaposition of subsequent A-RSs defining $W_\Delta$ using definition \ref{['eq:IS_short']} of the intermediate states $W_{{i - 1/2}, R}^*$ and $W_{{i + 1/2}, L}^*$.
• Figure 3: Graph of the function $\psi$ defined in \ref{['lem:def_psi']}. We observe that $\psi$ indeed satisfies properties \ref{['eq:psi_i']} -- \ref{['eq:psi_iv']}.
• Figure 4: Analytical solution \ref{['eq:exact_sol']} with $\varphi(x) = x$: $L^2$ errors and EOC for the first, second and third order schemes, with or without the high-order well-balanced correction.
• Figure 5: Perturbation of a hydrostatic equilibrium. Top panels: perturbation with initial amplitude $A=10^{-4}$. Bottom: perturbation with initial amplitude $A=10^{-12}$.

Theorems & Definitions (23)

• Definition 3.1: Interface Steady Solution (ISS)
• Definition 3.2: Well-balancedness
• Lemma 3.3
• proof
• Definition 3.4: Entropy stability
• Theorem 3.5
• proof
• Definition 3.6: Positivity preservation
• Lemma 3.7: Positivity of the pressure
• proof