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

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.

Figures (12)

• Figure 5.1: Example \ref{['ex:WBmoving']}: One-dimensional well-balancedness test with small pressure perturbation. The numerical solution of pressure perturbation and velocity perturbation of different flow regimes.
• Figure 5.2: Example \ref{['ex:Sod']}: One-dimensional Sod-like shock tube. The numerical solution at $T=0.4$ with $N=200$.
• Figure 5.3: Example \ref{['ex:Sod']}: One-dimensional Sod-like shock tube. The evolution of total entropy with time.
• Figure 5.4: Example \ref{['ex:Sod']}: One-dimensional Sod-like shock tube. The density at $T=0.4$ with different equilibrium states.
• Figure 5.5: Example \ref{['ex:DRF']}: One-dimensional double rarefaction wave problem. The numerical solution at $T=0.6$ with $N=800$.

Theorems & Definitions (35)

• Definition 3.1: Entropy conservative flux
• Definition 3.2: Entropy stable flux
• Theorem 3.1: Mass-conservation
• proof
• Theorem 3.2: Accuracy
• proof
• Remark 3.1
• Remark 3.2
• Theorem 3.3: Well-balancedness
• proof