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

We extend the entropy-stable oscillation-eliminating discontinuous Galerkin spectral element method (ES-OEDG) on curvilinear meshes \cite{YangFu26} to adaptive mesh refinement (AMR) grids with nonconforming interfaces. The formulation is developed for two-dimensional curvilinear quadrilateral meshes with 2:1 refinement and a single level of hanging nodes. A central contribution is the design and analysis of interface fluxes. We propose a flux-interpolation coupling that evaluates pairwise numerical fluxes between coarse- and fine-side nodal states before interpolation. The coupling is globally conservative and satisfies a semi-discrete entropy inequality under a scaled Guermond–Popov wave-speed condition. For $N\ge 2$, however, the sign correction required for entropy and positivity reduces formal consistency, and convergence on a severe curvilinear nonconforming mesh is close to first order. We also analyze the classical mortar flux, which retains high-order accuracy for smooth solutions but is not entropy stable and can exhibit entropy violations. Under suitable wave-speed and CFL conditions, both couplings preserve positive cell-average density and pressure, allowing the Zhang–Shu limiter to enforce nodal positivity. We further develop a complete AMR framework with an efficient indicator derived from the zero-order oscillation-elimination damping coefficient and an admissible refinement/coarsening transfer procedure that preserves conservation and positivity. Numerical experiments show that, despite reduced convergence rates on severe checkerboard meshes, the entropy-stable flux achieves robustness, solution quality, and resolution comparable to the mortar flux in shock-dominated AMR simulations.