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Author

Süleyman Yıldız

Bio: Süleyman Yıldız is an academic researcher from Middle East Technical University. The author has contributed to research in topics: Shallow water equations & Ordinary differential equation. The author has an hindex of 3, co-authored 12 publications receiving 20 citations.

Papers
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Journal ArticleDOI
TL;DR: In this paper, two different approaches for constructing reduced-order models (ROMs) for the two-dimensional shallow water equation (SWE) are presented, one based on the noncanonical Hamiltonian/Poisson form of the SWE, and the other based on a tensorial POD that preserves the linear-quadratic structure of SWE.
Abstract: 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/discrete empirical interpolation method (POD/DEIM) 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.

15 citations

Journal ArticleDOI
TL;DR: The efficiency of the proposed non‐intrusive method to construct reduced‐order models for NTSWE and compare it with an intrusive method (proper orthogonal decomposition) are illustrated and the predictive capabilities of both models outside the range of the training data are discussed.

11 citations

Journal ArticleDOI
TL;DR: Two different approaches for constructing reduced-order models (ROMs) for the two-dimensional shallow water equation (SWE) are presented and it is shown that in both approaches, the invariants of the SWE are preserved over a long period of time, leading to stable solutions.
Abstract: 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/discrete empirical interpolation method (POD/DEIM) 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.

8 citations

Journal ArticleDOI
TL;DR: This work presents a reduced-order model for a nonlinear cross-diffusion problem from population dynamics, for the Shigesada-Kawasaki-Teramoto (SKT) equation with Lotka-Volterra kinetics, and shows the decrease of the entropy numerically by the reduced solutions.

6 citations

Journal ArticleDOI
TL;DR: In this paper , a reduced-order model was developed for the rotating thermal shallow water equation (RTSWE) in the non-canonical Hamiltonian form with state-dependent Poisson matrix.
Abstract: In this paper, reduced-order models (ROMs) are developed for the rotating thermal shallow water equation (RTSWE) in the non-canonical Hamiltonian form with state-dependent Poisson matrix. The high fidelity full solutions are obtained by discretizing the RTSWE in space with skew-symmetric finite-differences, while preserving the Hamiltonian structure. The resulting skew-gradient system is integrated in time by the energy preserving average vector field (AVF) method. The ROM is constructed by applying proper orthogonal decomposition (POD) with the Galerkin projection, preserving the reduced skew-gradient structure, and integrating in time with the AVF method. The nonlinear terms of the Poisson matrix and Hamiltonian are approximated with the discrete empirical interpolation method (DEIM) to reduce the computational cost. The solutions of the resulting linear-quadratic reduced system is accelerated by the use of tensor techniques. The accuracy and computational efficiency of the ROMs are demonstrated for a numerical test problem. Preservation of the energy (Hamiltonian), and other conserved quantities, i.e., mass, buoyancy, and total vorticity show that the reduced-order solutions ensure the long-term stability of the solutions while exhibiting several orders of magnitude computational speedup over the full-order model (FOM). Furthermore, we show that the ROMs are able to accurately predict the test and training data, and capture the system behavior in the prediction phase.

5 citations


Cited by
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Journal Article
TL;DR: Reading simulating hamiltonian dynamics is a way as one of the collective books that gives many advantages and will greatly develop your experiences about everything.
Abstract: No wonder you activities are, reading will be always needed. It is not only to fulfil the duties that you need to finish in deadline time. Reading will encourage your mind and thoughts. Of course, reading will greatly develop your experiences about everything. Reading simulating hamiltonian dynamics is also a way as one of the collective books that gives many advantages. The advantages are not only for you, but for the other peoples with those meaningful benefits.

477 citations

Dissertation
01 Jan 2008
TL;DR: In this article, a method for rapid evaluation of flux-type outputs of interest from solutions to partial differential equations (PDEs) is presented within the reduced basis framework for linear, elliptic PDEs.
Abstract: A method for rapid evaluation of flux-type outputs of interest from solutions to partial differential equations (PDEs) is presented within the reduced basis framework for linear, elliptic PDEs. The central point is a Neumann-Dirichlet equivalence that allows for evaluation of the output through the bilinear form of the weak formulation of the PDE. Through a comprehensive example related to electrostatics, we consider multiple outputs, a posteriori error estimators and empirical interpolation treatment of the non-affine terms in the bilinear form. Together with the considered Neumann-Dirichlet equivalence, these methods allow for efficient and accurate numerical evaluation of a relationship mu->s(mu), where mu is a parameter vector that determines the geometry of the physical domain and s(mu) is the corresponding flux-type output matrix of interest. As a practical application, we lastly employ the rapid evaluation of s-> s(mu) in solving an inverse (parameter-estimation) problem.

116 citations

01 Jan 2013
TL;DR: In this paper, the authors investigated the process of pattern formation in a two dimensional domain for a reaction-diffusion system with nonlinear diffusion terms and the competitive Lotka-Volterra kinetics.
Abstract: Abstract In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.

105 citations