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

TL;DR

This paper introduces an entropy-stable high-order discontinuous Galerkin spectral element method for 2D Euler equations on curvilinear meshes, combining stability, oscillation control, and applicability to complex geometries.

Contribution

It extends the oscillation-eliminating DG method to general curvilinear meshes using projection operators, ensuring entropy stability and reducing computational cost.

Findings

Method achieves high accuracy on complex meshes.

Effectively suppresses nonphysical oscillations near shocks.

Demonstrates robustness and efficiency in numerical experiments.

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.