Invariant-domain-preserving limiting with Adaptive Mesh Refinement for Legendre–Gauss–Lobatto Discontinuous Galerkin Spectral Element Methods

Jul 16, 2026

Upload by Miles

Invariant-domain-preserving (IDP) limiting is developed for nonconforming interfaces in Legendre–Gauss–Lobatto Discontinuous Galerkin Spectral Element Methods (LGL-DGSEM) using adaptive mesh refinement on Cartesian meshes. The method extends convex limiting and graph-viscosity approaches to meshes with hanging nodes via conservative mortar formulation and low-order interface fluxes meeting IDP requirements. A sparsification strategy based on LGL subcell characteristic functions yields compact, conservative stencils that reduce to the conforming case and integrate with positivity limiting and shock-capturing for nonlinear hyperbolic conservation laws.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages30

Reading Time1 h 16 min

Introduction

High-order DGSEM and stability motivation

Spurious oscillations and invariant constraints

Stabilization background: AFC/FCT and invariant-domain frameworks

Goal: unified DGSEM + IDP limiting + AMR

Invariant-domain-preserving limiting with Adaptive Mesh Refinement for Legendre–Gauss–Lobatto Discontinuous Galerkin Spectral Element Methods Benjamin Bolma,∗ , Andrés M. Rueda-Ramírezb , Dmitri Kuzminc and Gregor J. Gassnera,d a Department of Mathematics and Computer Science, University of Cologne, Weyertal 86 -90, Cologne, 50931, NRW, Germany bETSIAE-UPM-School of Aeronautics, Universidad Politécnica de Madrid (UPM), Madrid, Spain c Institute of Applied Mathematics (LS III), TU Dortmund University, Vogelpothsweg 87, Dortmund, D-44227, NRW, Germany d Center for Data and Simulation Science, University of Cologne, Weyertal 86 -90, Cologne, 50931, NRW, Germany 13 Jul 2026 ARTICLE INFO Keywords: DG Spectral Element Method Shock Capturing Invariant domain preservation Adaptive Mesh Refinement Euler equations of gas dynamics AB STRACT 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. 1. Introduction High-order discontinuous Galerkin (DG) methods have become a popular framework for the numerical approximation of hyperbolic conservation laws due to their excellent accuracy, geometric flexibility, and suitability for modern high-performance computing architectures. Among these methods, the Discontinuous Galerkin Spectral Element Method (DGSEM) based on Legendre–Gauss–Lobatto (LGL) collocation points is particularly attractive because of its tensor-product structure, diagonal mass matrix, and efficient implementation on modern hardware. Moreover, splitform DGSEM discretizations possess favorable nonlinear stability properties and can be constructed to satisfy discrete entropy conservation or entropy stability for a broad class of hyperbolic systems [1–3] . Despite these advantages, high-order DG methods may generate spurious oscillations in the vicinity of discontinuities or under-resolved solution features. Such oscillations can violate essential physical constraints, including positivity of density and pressure for the compressible Euler equations or, more generally, invariant domain properties associated with the underlying system of conservation laws. Preserving these constraints is crucial for the robustness and reliability of simulations involving shocks, strong rarefactions, low-density regions, or under-resolved turbulence. The development of nonlinear stabilization techniques for high-order finite element and DG methods has a long history. A particularly influential class of approaches is provided by algebraic flux correction (AFC) and flux-corrected transport (FCT) methods, which combine a robust low-order discretization with a high-order target scheme through carefully designed antidiffusive fluxes an

FAQs

What problem does the paper address in LGL-DGSEM on nonconforming meshes?

It addresses preserving invariant-domain properties at nonconforming interfaces when using Legendre–Gauss–Lobatto DGSEM with adaptive mesh refinement, particularly on Cartesian meshes with hanging nodes.

How are invariant-domain-preserving interface fluxes constructed?

Starting from a conservative mortar formulation, the work derives low-order interface fluxes that satisfy the requirements of invariant-domain-preserving discretizations, while keeping the mortar fluxes conservative.

Why is sparsification used and what benefit does it provide?

A sparsification strategy based on LGL subcell characteristic functions avoids excessive diffusion from fully connected mortar couplings, producing compact interface stencils that still match the standard conforming formulation on matching interfaces.

Jul 16, 2026

Upload by Miles