Design Principle for Mode-Consistent Galerkin Closure under a Physical Energy Metric for Hyperbolic Systems
Abstract
This paper derives a design principle for structure-preserving Galerkin formulations of energy-conserving hyperbolic systems. The aim is to reproduce the modal-energy-exchange structure of the continuous system within a resolved finite-mode space. Total energy conservation follows from this structure. We introduce a state-dependent physical-energy metric H and derive the corresponding energy-compatibility identity. In the infinite-mode exact-integration model, the volume contribution has an antisymmetric representation after H-orthogonalization, yielding pairwise modal energy exchange. Interface contributions take the same exchange form. To reproduce this structure in the practical finite-mode system, we combine two constructions: a Galerkin projection coupled with the physical-energy metric that guarantees the H-metric summation-by-parts identity, and an energy-compatibility closure that removes the component of the compatibility action contributing to the scalar energy residual. With a shared numerical energy flux at interfaces, they close the total-energy balance of the finite-mode system while preserving pairwise modal energy exchange. We also compare the practical operator construction with the finite-mode exact-integration reference and obtain an O(h^p+1) defect estimate. Finally, we derive an equivalent form of the resulting equation in the fixed Galerkin basis for direct implementation.
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BibTeX
@misc{tomita2026design,
title = {{Design Principle for Mode-Consistent Galerkin Closure under a Physical Energy Metric for Hyperbolic Systems}},
author = {Hirofumi Tomita},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.05781},
eprint = {2607.05781},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.05781},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Design Principle for Mode-Consistent Galerkin Closure under a Physical Energy Metric for Hyperbolic Systems
AU - Hirofumi Tomita
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2607.05781
UR - https://arxiv.org/abs/2607.05781
AB - This paper derives a design principle for structure-preserving Galerkin formulations of energy-conserving hyperbolic systems. The aim is to reproduce the modal-energy-exchange structure of the continuous system within a resolved finite-mode space. Total energy conservation follows from this structure. We introduce a state-dependent physical-energy metric H and derive the corresponding energy-compatibility identity. In the infinite-mode exact-integration model, the volume contribution has an antisymmetric representation after H-orthogonalization, yielding pairwise modal energy exchange. Interface contributions take the same exchange form. To reproduce this structure in the practical finite-mode system, we combine two constructions: a Galerkin projection coupled with the physical-energy metric that guarantees the H-metric summation-by-parts identity, and an energy-compatibility closure that removes the component of the compatibility action contributing to the scalar energy residual. With a shared numerical energy flux at interfaces, they close the total-energy balance of the finite-mode system while preserving pairwise modal energy exchange. We also compare the practical operator construction with the finite-mode exact-integration reference and obtain an O(h^p+1) defect estimate. Finally, we derive an equivalent form of the resulting equation in the fixed Galerkin basis for direct implementation.
ER -