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

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.