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

Date

2021-10-01

Author

Uzunca, Murat
Karasözen, Bülent
Yıldız, Süleyman

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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.

Subject Keywords

Hamiltonian systems, Solitary waves, Kahan’s method, Energy preservation, Model order reduction, Tensor algebra

URI

https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85104447301&origin=inward
https://hdl.handle.net/11511/90331

Journal

Mathematics and Computers in Simulation

DOI

https://doi.org/10.1016/j.matcom.2021.03.042

Collections

Graduate School of Applied Mathematics, Article

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Citation Formats

M. Uzunca, B. Karasözen, and S. Yıldız, “Structure-preserving reduced-order modeling of Korteweg–de Vries equation,” Mathematics and Computers in Simulation, pp. 193–211, 2021, Accessed: 00, 2021. [Online]. Available: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85104447301&origin=inward.