Global energy preserving model reduction for multi-symplectic PDEs

TL;DR

This paper introduces a structure-preserving reduced-order modeling approach for multi-symplectic Hamiltonian PDEs that maintains global energy conservation, ensuring long-term stability and computational efficiency.

Contribution

The authors develop a ROM framework that preserves the discrete global energy of multi-symplectic PDEs using POD, Galerkin projection, and DEIM, with proven error bounds.

Findings

ROM accurately approximates solutions of KdV, ZK, and NLS equations.

Energy preservation ensures long-term stability of reduced solutions.

Method achieves computational efficiency in the online stage.

Abstract

Many Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced-order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are obtained by finite difference discretization in space and the global energy preserving average vector field (AVF) method. The ROM is constructed in the same way as the full-order model (FOM) applying proper orthogonal decomposition (POD) with the Galerkin projection. The reduced-order system has the same structure as the FOM, and preserves the discrete reduced global energy. Applying the discrete empirical interpolation method (DEIM), the reduced-order solutions are computed efficiently in the online stage. A priori error bound is derived for the DEIM approximation to the nonlinear Hamiltonian. The accuracy and computational efficiency of the ROMs are demonstrated for the Korteweg de Vries (KdV) equation, Zakharov-Kuznetzov (ZK) equation, and nonlinear Schr{\"o}dinger (NLS) equation in multi-symplectic form. Preservation of the reduced energies shows that the reduced-order solutions ensure the long-term stability of the solutions.