Global energy preserving model reduction for multi-symplectic PDEs


Uzunca M., KARASÖZEN B., Aydın A.

Applied Mathematics and Computation, vol.436, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 436
  • Publication Date: 2023
  • Doi Number: 10.1016/j.amc.2022.127483
  • Journal Name: Applied Mathematics and Computation
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, Public Affairs Index, zbMATH, Civil Engineering Abstracts
  • Keywords: discrete empirical interpolation method, energy preservation, Hamiltonian PDE, model reduction, multi-symplecticity, proper orthogonal decomposition
  • Middle East Technical University Affiliated: Yes

Abstract

© 2022 Elsevier Inc.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 Schrödinger (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.