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arXiv · 2604.21600

Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement

Jieling Yang·Guosheng Fu
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Abstract

We extend the entropy-stable oscillation-eliminating discontinuous Galerkin spectral element method (ES-OEDG) on curvilinear meshes to adaptive mesh refinement (AMR) grids with nonconforming interfaces. The formulation targets two-dimensional curvilinear quadrilateral meshes under a 2:1 refinement constraint, allowing a single level of hanging nodes. Elementwise volume discretization and geometric mapping are retained, while oscillation elimination and interface coupling are adapted for nonconforming interfaces. A central contribution is the design and analysis of numerical fluxes for such interfaces. We construct an entropy-stable flux that ensures global conservation and a semi-discrete entropy inequality. However, for polynomial degree N >= 2, negative entries in nonconforming interpolation operators lead to loss of formal high-order consistency. To address this, we propose a mortar-based flux that preserves high-order accuracy by interpolating at the solution level and evaluating standard two-point fluxes on fine-side mortars, at the cost of losing provable entropy stability. We also extend the Zhang--Shu positivity-preserving framework to curvilinear AMR meshes. Under forward Euler time stepping and a suitable CFL condition, the scheme using either flux preserves positivity of cell-average density and pressure. Combined with the Zhang--Shu limiter, this yields a fully discrete scheme maintaining admissibility at all nodal points. We further incorporate shock-indicator-based AMR and a conservative, positivity-preserving data transfer procedure between successive meshes, resulting in a robust and efficient algorithm. Numerical experiments on Cartesian and curvilinear AMR grids confirm high-order accuracy and robustness.

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Jieling Yang, Guosheng Fu. 2026-04-23. Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement. https://arxiv.org/abs/2604.21600

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