Structure-preserving reduced-order modeling of Korteweg–de Vries equation

Uzunca M., KARASÖZEN B., Yıldız S.

Mathematics and Computers in Simulation, vol.188, pp.193-211, 2021 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 188
  • Publication Date: 2021
  • Doi Number: 10.1016/j.matcom.2021.03.042
  • Journal Name: Mathematics and Computers in Simulation
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, Public Affairs Index, zbMATH
  • Page Numbers: pp.193-211
  • Keywords: Hamiltonian systems, Solitary waves, Kahan's method, Energy preservation, Model order reduction, Tensor algebra, NUMERICAL-SOLUTION, ZAKHAROV-KUZNETSOV, REDUCTION, SCHEMES, SYSTEMS, INTEGRATION, EVOLUTION, DYNAMICS, PDES
  • Middle East Technical University Affiliated: Yes


© 2021 International Association for Mathematics and Computers in Simulation (IMACS)Computationally efficient, structure-preserving reduced-order methods are developed for the Korteweg–de Vries (KdV) equations in Hamiltonian form. The semi-discretization in space by finite differences is based on the Hamiltonian structure. The resulting skew-gradient system of ordinary differential equations (ODEs) is integrated with the linearly implicit Kahan's method, which preserves the Hamiltonian approximately. We have shown, using proper orthogonal decomposition (POD), the Hamiltonian structure of the full-order model (FOM) is preserved by the reduced-order model (ROM). The reduced model has the same linear–quadratic structure as the FOM. The quadratic nonlinear terms of the KdV equations are evaluated efficiently by the use of tensorial framework, clearly separating the offline–online cost of the FOMs and ROMs. The accuracy of the reduced solutions, preservation of the conserved quantities, and computational speed-up gained by ROMs are demonstrated for the one-dimensional single and coupled KdV equations, and two-dimensional Zakharov–Kuznetsov equation with soliton solutions.