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

TL;DR

This paper introduces a conservative invariant-domain preserving projection method for hyperbolic systems with adaptive mesh refinement, ensuring physical property preservation and improved computational efficiency.

Contribution

It presents a new IDP projection technique that guarantees physical invariants in adaptive discretizations of hyperbolic systems, with rigorous proofs and practical implementation details.

Findings

Enhanced accuracy and efficiency demonstrated on benchmark problems

Method avoids ad hoc corrections for invariant preservation

Proven to be a reliable and rigorous approach for hyperbolic systems

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 \emph{ad hoc} corrections to preserve physical invariants.