Design Principle for Mode-Consistent Galerkin Closure under a Physical Energy Metric for Hyperbolic Systems

Hirofumi Tomita

arXiv:2607.05781v2 Primary: math.NA Secondary: physics.comp-ph

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.

Disclosure

“Acknowledgments. The author acknowledges partial support from JST/JICA SATREPS (Grant No. JPMJSA2109), thanks Drs. Yuta Kawai and Seiya Nishizawa for helpful discussions, and is grateful to his wife and daughter for their encourage- ment. Large language models were used for brainstorming, structural refinement, language editing, and preliminary organization of proof explanations. All mathemati- cal statements, derivations, proofs, technical decisions, and scientific conclusions were verified and”
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Structural counts

Pages 22 pdf

Theorems 4 source

Lemmas 15 source

Propositions 9 source

Corollaries 1 source

Definitions 15 source

Displayed equations 66 source

Bibliography entries 20 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.