Structure preserving model order reduction of shallow water equations


KARASÖZEN B., Yildiz S., UZUNCA M.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol.44, no.1, pp.476-492, 2021 (Peer-Reviewed Journal) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 44 Issue: 1
  • Publication Date: 2021
  • Doi Number: 10.1002/mma.6751
  • Journal Name: MATHEMATICAL METHODS IN THE APPLIED SCIENCES
  • Journal Indexes: Science Citation Index Expanded, Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.476-492
  • Keywords: discrete empirical interpolation, finite-difference methods, linearly implicit methods, preservation of invariants, proper orthogonal decomposition, tensorial proper orthogonal decomposition, FINITE-ELEMENT APPROXIMATIONS, MISSING POINT ESTIMATION, NONLINEAR MODEL, POTENTIAL-ENSTROPHY, INTERPOLATION METHOD, GENERAL-METHOD, ENERGY, SCHEMES, DECOMPOSITIONS, GALERKIN

Abstract

In this paper, we present two different approaches for constructing reduced-order models (ROMs) for the two-dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition(POD)/discrete empirical interpolation method that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit Kahan's method, and ROMs are constructed with the tensorial POD that preserves the linear-quadratic structure of the SWE. We show that in both approaches, the invariants of the SWE such as the energy, enstrophy, mass and circulation are preserved over a long period of time, leading to stable solutions. We conclude by demonstrating the accuracy and the computational efficiency of the reduced solutions by a numerical test problem.