A conservative invariant-domain preserving projection technique for hyperbolic systems under adaptive mesh refinement

2025/07/24 by Jake Harmon, Harmon, Jake, Martin Kronbichler +5
Engineering · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #Model Reduction and Neural Networks

paper · pdf · doi:10.48550/arxiv.2507.18717

Abstract

We propose a rigorous, conservative invariant-domain preserving (IDP) projection technique for hierarchical discretizations that enforces membership in physics-implied convex sets when mapping between solution spaces. When coupled with suitable refinement indicators, the proposed scheme enables a provably IDP adaptive numerical method for hyperbolic systems where preservation of physical properties is essential. In addition to proofs of these characteristics, we supply a detailed construction of the method in the context of a high-performance finite element code. To illustrate our proposed scheme, we study a suite of computationally challenging benchmark problems, demonstrating enhanced accuracy and efficiency properties while entirely avoiding ad hoc corrections to preserve physical invariants.

Citations

• Second-order invariant-domain preserving approximation to the multi-species Euler equations
• A high-order explicit Runge-Kutta approximation technique for the Shallow Water Equations
• Towards Adaptive Simulations of Dispersive Tsunami Propagation from an Asteroid Impact
• Second-order invariant domain preserving approximation of the Euler equations using convex limiting
• Invariant domains and first-order continuous finite element approximation for hyperbolic systems
• Invariant Domains and First-Order Continuous Finite Element Approximation for Hyperbolic Systems
• Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms

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