Linearly Implicit Global Energy Preserving Reduced-order Models for Cubic Hamiltonian Systems

TL;DR

This paper introduces a linearly implicit energy-preserving reduction method for large-scale cubic Hamiltonian PDEs, enabling accurate long-term simulations while maintaining energy conservation.

Contribution

It presents a novel linearly implicit global energy-preserving approach for reduced-order modeling of cubic Hamiltonian systems, ensuring energy conservation and long-time stability.

Findings

The method preserves global energy in reduced models.

It maintains local energy conservation laws in spatial discretization.

Numerical examples demonstrate improved long-term accuracy over classical methods.

Abstract

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.