An entropy-stable oscillation-eliminating dgsem for the euler equations on curvilinear meshes

TL;DR

The paper develops a high-order entropy-stable discontinuous Galerkin method (DGSEM) for the 2D compressible Euler equations on general curvilinear meshes by leveraging a summation-by-parts framework and discrete metric identities. It then integrates a modified oscillation-eliminating DG (OEDG) procedure, with a projection-based extension to curvilinear elements, using a shock-indicator to localize damping and reduce cost while preserving entropy stability and conservation. The resulting approach achieves accurate shockcapturing, preserves positivity, and demonstrates robust performance on Cartesian and curvilinear meshes across a range of challenging tests, including vortical, Riemann, and shock–shadow interactions. The work highlights a flexible framework that balances high-order accuracy, stability, and efficiency for complex geometries, with promising directions toward Navier–Stokes and MHD extensions.

Abstract

We develop an entropy-stable high-order numerical method for the two-dimensional compressible Euler equations on general curvilinear meshes. The proposed approach is based on a nodal discontinuous Galerkin spectral element method (DGSEM) that satisfies the summation-by-parts (SBP) property. At the semidiscrete level, entropy stability is established through the SBP structure and the discrete metric identities associated with curvilinear coordinate mappings. By incorporating entropy-stable numerical fluxes at element interfaces, a global discrete entropy inequality is obtained. To further control nonphysical oscillations near strong discontinuities, the entropy-stable DG formulation is combined with a modified oscillation-eliminating discontinuous Galerkin (OEDG) method, which was originally proposed in [59]. We observe that the zero-order damping coefficient in the original OEDG method naturally serves as an effective shock indicator, which enables localization of the oscillation control mechanism and significantly reduces computational cost. Moreover, while the original OEDG formulation relies on local orthogonal modal bases and is primarily restricted to simplicial meshes, we reformulate the OE procedure using projection operators, allowing for a systematic extension to general curvilinear meshes. The resulting method preserves conservation and entropy stability while effectively suppressing spurious oscillations. A series of challenging numerical experiments is presented to demonstrate the accuracy, robustness, and effectiveness of the proposed entropy-stable OEDG method on both Cartesian and curvilinear meshes.

Figures (10)

• Figure 1: Isentropic Euler vortex problem: initial condition of density and Curvilinear grids for Isentropic Euler Vortex problem. Mesh size: $N_x=N_y=M=40$
• Figure 2: Isentropic Euler vortex problem: convergency Rate of Isentropic Euler vortex problem in curvilinear mesh. Left figure has polynomial order of $N=3$, and the polynomial order of right figure is $N=4$
• Figure 3: Isentropic Euler vortex problem: how entropy integration changes with time. Left figure has polynomial order of $N=3$, and the polynomial order of right figure is $N=4$
• Figure 4: Two-dimensional Riemann problem 12: (a) the density profile at $t=0.2$; (b) the shock indicator profile at $t=0.2$.
• Figure 5: Two-dimensional Riemann problem 13: (a) the density profile at $t=0.3$; (b) the shock indicator profile at $t=0.3$.

Theorems & Definitions (17)

• Lemma 2.1: Properties of $D_{i,j}$
• Remark 2.1: Discrete metric identities
• Definition 3.1
• Lemma 3.1: Single-element analysis
• proof
• Theorem 3.2: Multi-element entropy analysis
• proof
• Remark 3.1: Entropy stability of Lax--Friedrichs flux
• Remark 3.2: Positivity preservation
• Remark 4.1: OE scale factor $s$