Structure-preserving nodal DG method for Euler equations with gravity II: general equilibrium states

TL;DR

This paper introduces an entropy-stable, well-balanced nodal DG scheme for Euler equations with gravity, capable of accurately preserving general equilibrium states including hydrostatic and moving equilibria.

Contribution

It presents a novel treatment of gravitational source terms combining entropy-conservative fluxes with entropy correction, ensuring structure preservation and positivity.

Findings

Scheme is entropy-stable and well-balanced for general equilibria.

Numerical experiments demonstrate robustness and efficiency.

Theoretical analysis confirms accuracy and structure-preserving properties.

Abstract

We develop an entropy-stable nodal discontinuous Galerkin (DG) scheme for the Euler equations with gravity, which is also well-balanced with respect to general equilibrium solutions, including both hydrostatic and moving equilibria. The core of our approach lies in a novel treatment of the gravitational source term, combining entropy-conservative numerical fluxes with a linear entropy correction. In addition, the proposed formulation is carefully designed to ensure compatibility with a positivity-preserving limiter. We provide a rigorous theoretical analysis to establish the accuracy and structure-preserving properties of the proposed scheme. Extensive numerical experiments confirm the robustness and efficiency of the scheme.