A Conservative Invariant-Domain Preserving Projection Technique for Hyperbolic Systems Under Adaptive Mesh Refinement

Abstract

Abstract. We propose a novel, rigorous, and conservative invariant-domain preserving (IDP) projection technique between solution spaces of differing discretizations. The projection preserves important physical invariants of the underlying hyperbolic system, for example, positivity of material density and internal energy for the compressible Euler equations. 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. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/conservation-laws/ryujin and in the supplementary materials ( ryujin-development.zip [938KB]). [Formula: see text]