This work discusses the model reduction problem for large-scale multi-symplectic PDEs with cubic invariants. For this, we present a linearly implicit global energy-preserving method to construct reduced-order models. This allows to construct reduced-order models in the form of Hamiltonian systems suitable for long-time integration. Furthermore, We prove that the constructed reduced-order models preserve global energy, and the spatially discrete equations also preserve the spatially-discrete local energy conversation law. We illustrate the efficiency of the proposed method using three numerical examples, namely a linear wave equation, the Korteweg-de Vries equation, and the Camassa-Holm equation, and present a comparison with the classical POD-Galerkin method.

翻译：本文研究了具有三次不变量的多辛偏微分方程的模型降阶问题。为此，我们提出了一种线性隐式的全局能量保持方法，用于构造降阶模型。该方法能够构建适用于长时间积分的哈密顿系统形式的降阶模型。此外，我们证明了所构造的降阶模型能够保持全局能量，且空间离散方程也满足空间局部能量守恒律。通过三个数值算例——线性波动方程、Korteweg-de Vries方程和Camassa-Holm方程，验证了所提方法的有效性，并与经典POD-Galerkin方法进行了对比。