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Öğe Intrusive and data-driven reduced order modelling of the rotating thermal shallow water equation(Elsevier Science Inc, 2022) Karasozen, Bulent; Yildiz, Suleyman; Uzunca, MuratIn this paper, we investigate projection-based intrusive and data-driven model order reduction in numerical simulation of rotating thermal shallow water equation (RTSWE) in parametric and non-parametric form. Discretization of the RTSWE in space with centered finite differences leads to Hamiltonian system of ordinary differential equations with linear and quadratic terms. The full-order model (FOM) is obtained by applying linearly implicit Kahan's method in time. Applying proper orthogonal decomposition with Galerkin projection (POD-G), we construct the intrusive reduced-order model (ROM). We apply operator inference (OpInf) with re-projection as data-driven ROM. In the parametric case, we make use of the parameter dependency at the level of the PDE without interpolating between the reduced operators. The least-squares problem of the OpInf is regularized with the minimum norm solution. Both ROMs behave similarly and are able to accurately predict the in the test and training data and capture system behaviour in the prediction phase with several orders of magnitude in computational speed-up over the FOM. The preservation of system physics such as the conserved quantities of the RTSWE by both ROMs enable that the models fit better to data and stable solutions are obtained in long-term predictions which are robust to parameter changes. (C) 2022 Elsevier Inc. All rights reserved.Öğe Reduced order modelling of nonlinear cross-diffusion systems(Elsevier Science Inc, 2021) Karasozen, Bulent; Mulayim, Gulden; Uzunca, Murat; Yildiz, SuleymanIn this work, we present reduced-order models (ROMs) for a nonlinear cross-diffusion problem from population dynamics, the Shigesada-Kawasaki-Teramoto (SKT) equation with Lotka-Volterra kinetics. The formation of the patterns of the SKT equation consists of a fast transient phase and a long stationary phase. Reduced order solutions are computed by separating the time into two time-intervals. In numerical experiments, we show for one- and two-dimensional SKT equations with pattern formation, the reduced-order solutions obtained in the time-windowed form, i.e., principal decomposition framework, are more accurate than the global proper orthogonal decomposition solutions obtained in the whole time interval. The finite-difference discretization of the SKT equation in space results in a system of linear-quadratic ordinary differential equations. The ROMs have the same linear-quadratic structure as the full order model. Using the linear-quadratic structure of the ROMs, the computation of the reduced-order solutions is further accelerated by the use of proper orthogonal decomposition in a tensorial framework so that the computations in the reduced system are independent of the full-order solutions. Furthermore, the prediction capabilities of the ROMs are illustrated for one- and two-dimensional patterns. Finally, we show that the entropy is decreasing by the reduced solutions, which is important for the global existence of solutions to the nonlinear cross-diffusion equations such as the SKT equation. (C) 2021 Elsevier Inc. All rights reserved.Öğe Structure preserving model order reduction of shallow water equations(Wiley, 2021) Karasozen, Bulent; Yildiz, Suleyman; Uzunca, MuratIn 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.Öğe Structure-preserving reduced-order modeling of Korteweg-de Vries equation(Elsevier, 2021) Uzunca, Murat; Karasozen, Bulent; Yildiz, SuleymanComputationally 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 (EOM) 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. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
