Global energy preserving model reduction for multi-symplectic PDEs

Date

2023-01-01

Author

Uzunca, Murat
Karasözen, Bülent
Aydın, Ayhan

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

169
views

0
downloads

© 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.

Subject Keywords

discrete empirical interpolation method, energy preservation, Hamiltonian PDE, model reduction, multi-symplecticity, proper orthogonal decomposition

URI

https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85136584284&origin=inward
https://hdl.handle.net/11511/100351

Journal

Applied Mathematics and Computation

DOI

https://doi.org/10.1016/j.amc.2022.127483

Collections

Department of Mathematics, Article

Suggestions

OpenMETU
Core

Structure preserving model order reduction of shallow water equations Karasözen, Bülent; UZUNCA, MURAT (2020-07-01) 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 eq...
Model order reduction for nonlinear Schrodinger equation Karasözen, Bülent; Uzunca, Murat (2015-05-01) We apply the proper orthogonal decomposition (POD) to the nonlinear Schrodinger (NLS) equation to derive a reduced order model. The NLS equation is discretized in space by finite differences and is solved in time by structure preserving symplectic mid-point rule. A priori error estimates are derived for the POD reduced dynamical system. Numerical results for one and two dimensional NLS equations, coupled NLS equation with soliton solutions show that the low-dimensional approximations obtained by POD reprodu...
Structure preserving reduced order modeling for gradient systems Akman Yildiz, Tugba; UZUNCA, MURAT; Karasözen, Bülent (2019-04-15) Minimization of energy in gradient systems leads to formation of oscillatory and Turing patterns in reaction-diffusion systems. These patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state. In this paper, a reduced order model (ROM) is developed which preserves the gradient dissipative structure. The coupled system of reaction-diffusion equations are discretized in space by the symmetric interior penalty discontinuous...
Energy preserving model order reduction of the nonlinear Schrodinger equation Karasözen, Bülent (2018-12-01) An energy preserving reduced order model is developed for two dimensional nonlinear Schrodinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orth...
Polynomial Multiplication over Binary Fields Using Charlier Polynomial Representation with Low Space Complexity AKLEYLEK, SEDAT; Cenk, Murat; Özbudak, Ferruh (2010-12-15) In this paper, we give a new way to represent certain finite fields GF(2(n)). This representation is based on Charlier polynomials. We show that multiplication in Charlier polynomial representation can be performed with subquadratic space complexity. One can obtain binomial or trinomial irreducible polynomials in Charlier polynomial representation which allows us faster modular reduction over binary fields when there is no desirable such low weight irreducible polynomial in other representations. This repre...

Citation Formats

M. Uzunca, B. Karasözen, and A. Aydın, “Global energy preserving model reduction for multi-symplectic PDEs,” Applied Mathematics and Computation, vol. 436, pp. 0–0, 2023, Accessed: 00, 2022. [Online]. Available: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=85136584284&origin=inward.