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

Abstract

This paper derives a design principle for Galerkin approximations of energy-conserving hyperbolic systems, following Arakawa's philosophy of structure preservation. The aim is to reproduce, within a resolved finite-mode space, the modal-energy-exchange structure of the continuous system, so that total energy conservation follows as a consequence. We introduce a state-dependent metric H(U) representing the physical energy density and derive the corresponding energy-compatibility identity. In the exact-integration infinite-mode reference model, H-orthogonalization makes the volume operator antisymmetric, so the modal energy balance is expressed as pairwise exchange between modes. Boundary and interface contributions are likewise represented as exchanges with adjacent-element modes, with internal exchanges cancelling pairwise. To reproduce this structure in a semi-discrete finite-mode system, we combine two constructions: a Galerkin projection coupled with the physical energy metric, which guarantees the H-metric summation-by-parts identity, and an energy-compatibility closure, which cancels the compatibility residual by modifying the evolution of the H-metric mass matrix. The resulting finite-mode system recovers the modal-energy-exchange structure. For discontinuous element-boundary traces, the interface contribution is closed by a shared numerical energy flux satisfying the same pairwise balance. We also compare the practical operator construction with the exact-integration finite-mode reference model. The defect in the antisymmetric modal-energy-exchange operator is decomposed into fixed-quadrature and projection-quadrature contributions, yielding an O(h^{p+1})-consistent estimate. Finally, transformation back to the original Galerkin basis gives an equivalent fixed-basis coefficient equation that is directly implementable.