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

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  • Uzunca, Murat
  • Karasözen, Bülent
  • Yıldız, Süleyman

Abstract

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.

Suggested Citation

  • Uzunca, Murat & Karasözen, Bülent & Yıldız, Süleyman, 2021. "Structure-preserving reduced-order modeling of Korteweg–de Vries equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 193-211.
    • Handle: RePEc:eee:matcom:v:188:y:2021:i:c:p:193-211
    DOI: 10.1016/j.matcom.2021.03.042
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    References listed on IDEAS

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    1. Timo Reis & Tatjana Stykel, 2007. "Stability analysis and model order reduction of coupled systems," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 13(5), pages 413-436, October.
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