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

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.

💡 Research Summary

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The paper presents a novel high‑order discontinuous Galerkin spectral element method (DGSEM) for the two‑dimensional compressible Euler equations on general curvilinear meshes that simultaneously guarantees entropy stability and eliminates non‑physical oscillations near strong discontinuities. The authors build on the nodal DGSEM based on Legendre–Gauss–Lobatto (LGL) points, which possesses a summation‑by‑parts (SBP) property. By carefully discretizing the geometric terms of the curvilinear mapping, they enforce the discrete metric identities, ensuring that the transformed system on the reference element respects both free‑stream preservation and the entropy inequality at the semi‑discrete level. Entropy‑stable numerical fluxes are employed at element interfaces, yielding a global discrete entropy inequality.

To address the well‑known Gibbs phenomenon that plagues high‑order schemes near shocks, the authors incorporate the oscillation‑eliminating discontinuous Galerkin (OEDG) technique. The original OEDG, introduced in a previous work, formulates the damping mechanism as a local pseudo‑time evolution problem that can be solved exactly, thereby avoiding the stiffness associated with artificial viscosity. However, the original formulation was limited to orthogonal modal bases on simplicial meshes. In this work, the OEDG procedure is reformulated using projection operators, which makes it applicable to any curvilinear quadrilateral element. Crucially, the zero‑order damping coefficient that appears in the OEDG algorithm is shown to act as an effective shock indicator, closely related to the classical KXRCF sensor. By activating the OEDG damping only in cells where this coefficient exceeds a prescribed threshold, the method localizes the oscillation‑control mechanism and dramatically reduces computational cost.

The combined entropy‑stable OEDG method retains conservation, satisfies the discrete entropy inequality, and suppresses spurious oscillations without sacrificing high‑order accuracy. A series of demanding numerical tests are presented, including smooth vortex propagation on both Cartesian and distorted meshes, shock‑vortex interaction, strong shock reflection, and high‑Mach number flows. In all cases the method demonstrates robust shock capturing, accurate shock speeds, and preservation of positivity for density and pressure. Moreover, the entropy production remains bounded by the theoretical entropy‑stable limit, confirming the analytical results.

Overall, the paper makes three significant contributions: (1) a rigorous entropy‑stable DGSEM framework on curvilinear meshes based on SBP operators and discrete metric identities; (2) a systematic extension of the OEDG shock‑capturing technique to general curvilinear geometries via projection‑based formulation; and (3) the identification of the OEDG zero‑order damping coefficient as a cheap, reliable shock sensor that enables selective application of the oscillation‑eliminating mechanism. This work therefore provides a powerful, high‑order, entropy‑consistent tool for simulating compressible flows with complex geometry and strong discontinuities.