• Mathematics > Numerical Analysis [Submitted on 17 Feb 2026] Title:An entropy-stable oscillation-eliminating dgsem for the euler equations on curvilinear meshes View PDF HTML (experimental)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.
Article Summaries:
- A new high‑order numerical scheme has been proposed for solving the two‑dimensional compressible Euler equations on general curvilinear meshes. The method is a nodal discontinuous Galerkin spectral element method (DGSEM) that satisfies a summation‑by‑parts (SBP) property, ensuring entropy stability at the semidiscrete level through discrete metric identities. Entropy‑stable numerical fluxes at element interfaces yield a global discrete entropy inequality. To suppress nonphysical oscillations near shocks, the DGSEM is coupled with a modified oscillation‑eliminating DG (OEDG) approach; the zero‑order damping term acts as an effective shock indicator. By reformulating the OE procedure with projection operators, the scheme extends beyond simplicial meshes while preserving conservation and entropy stability. Numerical tests on Cartesian and curvilinear grids confirm its accuracy and robustness.
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