Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes
Abstract
This paper presents a unified invariant-domain-preserving (IDP) framework for hybrid discretizations of hyperbolic conservation laws, including active flux and PAMPA methods, in which cell averages are updated conservatively while cell-boundary point values evolve under a possibly non-conservative operator. The main challenge is to preserve admissibility for these two coupled sets of states under a single local stability condition, without adding evolved degrees of freedom or relying on post-update repairs. For the point-value update, we introduce an admissibility transform based on a barrier--Legendre map for convex admissible interiors described by concave constraints, and prove that the inverse map is globally defined and Lipschitz continuous on the relevant sets. For the conservative cell-average update, we establish a structural obstruction theorem: the single-state continuous physical flux built from admissible boundary traces alone cannot provide a conservative IDP guarantee, for any prescribed CFL number, when internal reconstruction values are uncontrolled. This identifies the missing local control that must be supplied by an additional IDP flux mechanism. To realize this mechanism explicitly, we combine cell average decompositions (CAD), geometric quasilinearization, and local a priori scaling to construct admissible, generally discontinuous trace states before flux evaluation. Under an explicit trace-based CFL condition, with constants determined by local CAD weights and trace-state wave-speed bounds, the coupled hybrid update preserves the prescribed invariant domain. Concrete third-order schemes are developed on triangular, Cartesian, convex quadrilateral, and general convex polygonal meshes. Numerical results demonstrate the designed order of accuracy for smooth solutions and the strict preservation of physical admissibility.
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BibTeX
@misc{ding2026invariant,
title = {{Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes}},
author = {Shengrong Ding and Shumo Cui and Remi Abgrall and Kailiang Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10476},
eprint = {2609.10476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10476},
}
RIS
TY - UNPB
TI - Invariant domain preservation for hybrid point-value and cell-average discretizations of hyperbolic equations on general meshes
AU - Shengrong Ding
AU - Shumo Cui
AU - Remi Abgrall
AU - Kailiang Wu
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10476
UR - https://arxiv.org/abs/2609.10476
AB - This paper presents a unified invariant-domain-preserving (IDP) framework for hybrid discretizations of hyperbolic conservation laws, including active flux and PAMPA methods, in which cell averages are updated conservatively while cell-boundary point values evolve under a possibly non-conservative operator. The main challenge is to preserve admissibility for these two coupled sets of states under a single local stability condition, without adding evolved degrees of freedom or relying on post-update repairs. For the point-value update, we introduce an admissibility transform based on a barrier--Legendre map for convex admissible interiors described by concave constraints, and prove that the inverse map is globally defined and Lipschitz continuous on the relevant sets. For the conservative cell-average update, we establish a structural obstruction theorem: the single-state continuous physical flux built from admissible boundary traces alone cannot provide a conservative IDP guarantee, for any prescribed CFL number, when internal reconstruction values are uncontrolled. This identifies the missing local control that must be supplied by an additional IDP flux mechanism. To realize this mechanism explicitly, we combine cell average decompositions (CAD), geometric quasilinearization, and local a priori scaling to construct admissible, generally discontinuous trace states before flux evaluation. Under an explicit trace-based CFL condition, with constants determined by local CAD weights and trace-state wave-speed bounds, the coupled hybrid update preserves the prescribed invariant domain. Concrete third-order schemes are developed on triangular, Cartesian, convex quadrilateral, and general convex polygonal meshes. Numerical results demonstrate the designed order of accuracy for smooth solutions and the strict preservation of physical admissibility.
ER -