h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes

Abstract

High-order methods offer superior dispersion and dissipation properties compared to low-order schemes but require robust stabilization for discontinuities. To ensure stability, local artificial viscosity is common, but often degrades sub-element resolution. Conversely, subcell resolution preserving limiting strategies such as the finite volume subcell method are typically restricted to uniform topologies, such as purely hexahedral or simplex meshes, or linear elements. This leaves a significant gap in treating the hybrid-element topologies necessary for complex engineering geometries. To bridge this gap, we introduce a robust shock-capturing approach for the discontinuous Galerkin spectral element method on mixed curvilinear meshes containing hexahedral, prismatic, tetrahedral, and pyramid elements. Non-hexahedral elements are handled via collapsed coordinate transformations. The proposed method utilizes an $h$-adaptive finite volume subcell scheme with an arbitrary subcell resolution up to $2\mathcal{N}+1$. Special care is taken to ensure conforming subcell distributions across different element types. Crucially, discrete conservation is proven theoretically and verified numerically, alongside validations for high-order convergence and robust shock capturing. Finally, the method's applicability to complex configurations is demonstrated through a simulation of the flow around a NACA 0012 airfoil.