Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement
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| Format: | Preprint |
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2026
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| author | Yang, Jieling Fu, Guosheng |
| author_facet | Yang, Jieling Fu, Guosheng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_21600 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement Yang, Jieling Fu, Guosheng Numerical Analysis 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. |
| title | Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2604.21600 |