发表机构
RIKEN Center for Computational Science（理化学研究所计算科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究双曲系统伽辽金近似设计原理，基于荒川理念，通过引入物理能量密度度量\(H(U)\)，结合伽辽金投影与能量相容性封闭构造，使有限模式系统恢复模式能量交换结构，给出相关估计与方程。
AI 中文摘要
本文遵循荒川结构保持理念，推导了能量守恒双曲系统的伽辽金近似设计原理。旨在在解析有限模式空间中重现连续系统的模式能量交换结构，以实现总能量守恒。引入表示物理能量密度的状态依赖度量\(H(U)\)并推导相应的能量相容性恒等式。在精确积分无限模式参考模型中，\(H\)正交化使体积算子反对称，模式能量平衡表现为模式间成对交换。为在半离散有限模式系统中重现此结构，结合伽辽金投影与物理能量度量及能量相容性封闭两种构造。所得有限模式系统恢复了模式能量交换结构，对于不连续单元边界迹线，界面贡献由满足相同成对平衡的共享数值能量通量封闭。还将实际算子构造与精确积分有限模式参考模型比较，将反对称模式能量交换算子缺陷分解为固定求积和投影求积贡献，得到\(O(h^{p + 1})\)一致估计，最后转换回原始伽辽金基给出可直接实现的等效固定基系数方程。
英文摘要
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.
Comments23 pages, 2 figures. Minor revisions and clarifications, including the interface formulation, normal orientation and sign conventions, and notation. The principal results are unchanged
