Global energy preserving model reduction for multi-symplectic PDEs

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{ö}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.

References in corpus (9)

• A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions
• Structure Preserving Model Reduction of Parametric Hamiltonian Systems
• Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics
• Structure-Preserving Galerkin POD Reduced-Order Modeling of Hamiltonian Systems
• General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs
• Intrusive and non-intrusive reduced order modeling of the rotating thermal shallow water equation
• Energy preserving model order reduction of the nonlinear Schrödinger equation
• Structure Preserving Model Order Reduction of Shallow Water Equations
• Energy preserving reduced-order modelling of thermal shallow water equation