Structure-preserving nodal DG method for the Euler equations with gravity: well-balanced, entropy stable, and positivity preserving

TL;DR

The paper develops a structure-preserving nodal DG method for the Euler equations with gravity that is well-balanced, entropy-stable, and positivity-preserving. By aligning a discretized source term with entropy-conservative fluxes within a Gauss–Lobatto SBP framework, the scheme exactly preserves hydrostatic equilibria while dissipating entropy in accordance with the second law. A positivity-preserving scaling limiter ensures positive density and pressure under a CFL condition, and the approach extends from 1D to 2D with analogous properties. Numerical experiments in both dimensions demonstrate robust accuracy, the ability to resolve small perturbations about equilibria, and stable shock capturing without excessive limiting. The method offers a theoretically justified, high-order tool for simulations of astrophysical and atmospheric flows where gravity and thermodynamic realism are essential.

Abstract

We propose an entropy stable and positivity preserving discontinuous Galerkin (DG) scheme for the Euler equations with gravity, which is also well-balanced for hydrostatic equilibrium states. To achieve these properties, we utilize the nodal DG framework and carefully design the source term discretization using entropy conservative fluxes. Furthermore, we demonstrate that the proposed methodology is compatible with a positivity preserving scaling limiter, ensuring positivity of density and pressure under an appropriate CFL condition. To the best of our knowledge, this is the first DG scheme to simultaneously achieve these three properties with theoretical justification. Numerical examples further demonstrate its robustness and efficiency.

Figures (9)

• Figure 5.1: Example 5.1: One-dimensional well-balancedness test with small velocity perturbation. The numerical solution at $T = 1.5$ on $N=200$ meshes.
• Figure 5.2: Example 5.2: One-dimensional Sod-like shock tube. The numerical solution at $T = 0.4$ on $N=200$ meshes.
• Figure 5.3: Example 5.2: One-dimensional Sod-like shock tube. The evolution of total entropy on $N=200$ meshes.
• Figure 5.4: Example 5.3: One-dimensional double rarefaction problem. The numerical solution at $T = 0.6$ on $N=800$ meshes.
• Figure 5.5: Example 5.4: Two-dimensional well-balancedness test. The numerical solution at $T = 0.15$ on $N_x\times N_y=100\times 100$ meshes. 15 contour lines are used.

Theorems & Definitions (20)

• Lemma 3.1
• Lemma 3.2
• Definition 3.1
• Definition 3.2
• Theorem 3.1: Well-balancedness
• proof
• Remark 3.1
• Remark 3.2
• Theorem 3.2: Entropy stability
• proof