On the involution preserving properties of semi-discrete Active Flux methods on simplex
Abstract
We study the involution-preserving properties of the Active Flux method on unstructured triangular meshes. Two model problems are considered: the two-dimensional linear acoustic system, which satisfies a curl involution on the velocity field, and the two-dimensional Maxwell equations with an in-plane magnetic field, which satisfy a divergence involution. Both systems can be written in conservative form. The question we address is how a single discretization can satisfy the conservative form and the involution at the same time. Within a discrete Active Flux framework on simplices, we derive explicit formulas for a discrete curl operator that is exactly preserved by the scheme for acoustics, and for a discrete divergence operator preserved for the Maxwell system. The analysis applies to general simplex meshes and to any polynomial degree, including extensions of the method beyond third order. Numerical experiments with the third-order scheme confirm the theoretical results. Highlights: 1-Active Flux on unstructured triangular meshes preserves discrete involutions, 2-Explicit discrete curl operator preserved by the scheme for linear acoustics, 3- Explicit discrete divergence operator preserved for 2D Maxwell equations, 4- Results hold on general simplex meshes,% and for any polynomial degree 5- Third-order numerical experiments confirm the theoretical results. \end{itemize}
Related
- Involution-preserving ring isomorphisms in norm between unital C*-algebras
- Involution-Preserving schemes through constrained mesh motion: a novel Arbitrary-Lagrangian-Eulerian framework
- A Network Simplex Framework for Discrete Unbalanced Optimal Transport
- Discrete Unique Continuation on Simplex
- Admissibility-Preserving Constraint Lifting for Conditional Deadbeat Control of Discrete-Time Nonlinear Systems
BibTeX
@misc{abgrall2026involution,
title = {{On the involution preserving properties of semi-discrete Active Flux methods on simplex}},
author = {Remi Abgrall and Yongle Liu and Mario Ricchiuto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04349},
eprint = {2610.04349},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04349},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the involution preserving properties of semi-discrete Active Flux methods on simplex
AU - Remi Abgrall
AU - Yongle Liu
AU - Mario Ricchiuto
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04349
UR - https://arxiv.org/abs/2610.04349
AB - We study the involution-preserving properties of the Active Flux method on unstructured triangular meshes. Two model problems are considered: the two-dimensional linear acoustic system, which satisfies a curl involution on the velocity field, and the two-dimensional Maxwell equations with an in-plane magnetic field, which satisfy a divergence involution. Both systems can be written in conservative form. The question we address is how a single discretization can satisfy the conservative form and the involution at the same time. Within a discrete Active Flux framework on simplices, we derive explicit formulas for a discrete curl operator that is exactly preserved by the scheme for acoustics, and for a discrete divergence operator preserved for the Maxwell system. The analysis applies to general simplex meshes and to any polynomial degree, including extensions of the method beyond third order. Numerical experiments with the third-order scheme confirm the theoretical results. Highlights: 1-Active Flux on unstructured triangular meshes preserves discrete involutions, 2-Explicit discrete curl operator preserved by the scheme for linear acoustics, 3- Explicit discrete divergence operator preserved for 2D Maxwell equations, 4- Results hold on general simplex meshes,% and for any polynomial degree 5- Third-order numerical experiments confirm the theoretical results. \end{itemize}
ER -