We derived an invariant-domain-preserving first-order approach for nonconforming interfaces (mortar flux). It is an extension to the subcell convex limiting approach used before for conforming meshes. With that, we can simulate challenging problems, guaranteeing numerical and physical stability, and use the efficiency advantages of AMR.

In the following example, we simulated a Mach 2000 astrophysical Jet with the 2D compressible Euler equations by combining a forth-order DGSEM with a first-order subcell FV scheme and applying physical admissibility by ensuring positivity of density and pressure.
At end time T=0.0015, we have 79,708 elements, i.e. 1,275,328 degrees of freedom. We use the entropy-conserving and kinetic energy preserving flux of Chandrashekar for the volume fluxes and the local Lax-Friedrichs flux for the surface fluxes of the DG and FV methods.

Abstract of the preprint:
We present an invariant-domain-preserving (IDP) treatment of nonconforming interfaces for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral Element Methods (LGL-DGSEM) with adaptive mesh refinement (AMR) on Cartesian meshes. The proposed methodology extends recently developed convex limiting and graph-viscosity frameworks for DGSEM to meshes containing hanging nodes.
Starting from a conservative mortar formulation, we derive low-order interface fluxes that satisfy the requirements of invariant-domain-preserving discretizations. To avoid the excessive diffusion associated with fully connected mortar couplings, a sparsification strategy based on LGL subcell characteristic functions is introduced, yielding compact interface stencils. The resulting mortar fluxes remain conservative, reduce to the standard conforming formulation on matching interfaces, and naturally fit into graph-viscosity-based low-order schemes used for convex limiting.
The proposed construction provides the missing ingredient required to combine high-order DGSEM discretizations, invariant-domain-preserving limiting, and adaptive mesh refinement within a unified framework for nonlinear hyperbolic conservation laws. We provide numerical verifications of the properties of the proposed scheme and run challenging simulations that require positivity limiting and shock-capturing.

Link to the paper: https://arxiv.org/abs/2607.06045

Further References:

•  A. M. Rueda-Ramírez, W. Pazner, G. J. Gassner, Subcell limiting strategies for discontinuous Galerkin spectral element methods, Computers & Fluids 247 (2022) 105627.
• A. M. Rueda-Ramírez, B. Bolm, D. Kuzmin, G. J. Gassner, Monolithic convex limiting for legendre-gauss-lobatto discontinuous galerkin spectral-element methods, Communications on Applied Mathematics and Computation (2024)