Piece-wise Symplectic Model Reduction on Quadratically Embedded Manifolds

In this work, we present a piece-wise symplectic model order reduction (MOR) method for Hamiltonian systems on quadratically embedded manifolds. For Hamiltonian systems, which suffer from slowly decaying Kolmogorov 𝑁-widths, linear-subspace reduced-order models (ROMs) of low dimension can have insufficient accuracy. The recently proposed symplectic manifold Galerkin projection combined with the quadratic manifold cotangent lift approximation (QMCL) is a symplectic MOR method that can achieve higher accuracy than linear-subspace symplectic MOR methods. In this paper, we improve the online computational complexity and energy-preserving ability of the QMCL by proposing a piece-wise symplectic MOR approach. First, the QMCL map is approximated by a linear symplectic map on each discrete time-interval. Then, the symplectic Galerkin projection is applied to obtain a sequence of reduced-order Hamiltonian systems. In case that the Hamiltonian of the full-order model is a polynomial, the sequence of the Hamiltonians of the ROMs can be preserved up to a multiple of a pre-given tolerance used in the Newton iteration. In the numerical example, we investigate the approximation quality and the energy-preservation of the proposed algorithm.