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Research Project Lineer Olmaya Üçlü Schrödinger Denklemi için Yapı Koruyan Sayısal Yöntemler(2016) Aydın, Ayhan; Ertuğ, Sevim-Article Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions(Computer Physics Communications, 2007) Aydın, Ayhan; Karasözen, BülentWe consider for the integration of coupled nonlinear Schrödinger equations with periodic plane wave solutions a splitting method from the class of symplectic integrators and the multi-symplectic six-point scheme which is equivalent to the Preissman scheme. The numerical experiments show that both methods preserve very well the mass, energy and momentum in long-time evolution. The local errors in the energy are computed according to the discretizations in time and space for both methods. Due to its local nature, the multi-symplectic six-point scheme preserves the local invariants more accurately than the symplectic splitting method, but the global errors for conservation laws are almost the same.Article Lobatto Iiia–iiib Discretization of the Strongly Coupled Nonlinear Schrödinger Equation(Journal of Computational and Applied Mathematics, 2009) Aydın, Ayhan; Karasözen, BülentIn this paper, we construct a second order semi-explicit multi-symplectic integrator for the strongly coupled nonlinear Schrödinger equation based on the two-stage Lobatto IIIA–IIIB partitioned Runge–Kutta method. Numerical results for different solitary wave solutions including elastic and inelastic collisions, fusion of two solitons and with periodic solutions confirm the excellent long time behavior of the multi-symplectic integrator by preserving global energy, momentum and mass.Publication Semi-Explicit Multi-Symplectic Integration of Nonlinear Schrodinger Equation(2015) Aydın, AyhanIn this paper we apply Lobatto IIIA-IIIB type multi-symplectic discretization in space and time to the nonlinear Schrödinger equation. The resulting scheme is semi-explicit in time and therefore more efficient than implicit multisymplectic schemes. Numerical results confirm excellent long time conservation of the local and global conserved quantities like the energy, momentum and norm.Article Citation - WoS: 6Citation - Scopus: 6Exact and Nonstandard Finite Difference Schemes for the Burgers Equation B(2, 2)(Tubitak Scientific & Technological Research Council Turkey, 2021) Köroğlu, Canan; Aydın, AyhanIn this paper, we consider the Burgers equation B(2, 2) . Exact and nonstandard finite difference schemes(NSFD) for the Burgers equation B(2, 2) are designed. First, two exact finite difference schemes for the Burgers equationB(2, 2) are proposed using traveling wave solution. Then, two NSFD schemes are represented for this equation. Thesetwo NSFD schemes are compared with a standard finite difference (SFD) scheme. Numerical results show that the NSFDschemes are accurate and efficient in the numerical simulation of the kink-wave solution of the B(2, 2) equation. We seethat although the SFD scheme yields numerical instability for large step sizes, NSFD schemes provide reliable results forlong time integration. Local truncation errors show that the NSFD schemes are consistent with the B(2, 2) equation.Book Part Multisymplectic Integrators for Coupled Nonlinear Partial Differential Equations(Nova Science Publishers, Inc., 2012) Karas̈ozen,B.; Aydın, Ayhan; Aydin,A.; Aydın, Ayhan; Mathematics; MathematicsThe numerical solution of nonlinear partial differential equations (PDEs) using symplectic geometric integrators has been the subject of many studies in recent years. Many nonlinear partial differential equations can be formulated as an infinite dimensional Hamiltonian system. After semi-discretization in the space variable, a system of Hamiltonian ordinary differential equations (ODEs) is obtained, for which various symplectic integrators can be applied. Numerical results show that symplectic schemes have superior performance, especially in long time simulations. The concept of multisymplectic PDEs and multisymplectic schemes can be viewed as the generalization of symplectic schemes. In the last decade, many multisymplectic methods have been proposed and applied to nonlinear PDEs, like to nonlinear wave equation, nonlinear Schr̈odinger equation, Korteweg de Vries equation, Dirac equation, Maxwell equation and sine-Gordon equation. In this review article, recent results of multisymplectic integration on the coupled nonlinear PDEs, the coupled nonlinear Schr̈odinger equation, the modified complex Korteweg de Vries equation and the Zakharov system will be given. The numerical results are discussed with respect to the stability of the schemes, accuracy of the solutions, conservation of the energy and momentum, preservation of dispersion relations. © 2012 Nova Science Publishers, Inc. All rights reserved.Article An Unconventional Splitting for Korteweg de Vries–Burgers Equation(EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS, 2015) Aydın, AyhanNumerical solutions of the Korteweg de Vries–Burgers (KdVB) equation based on splitting is studied. We put a real parameter into a KdVB equation and split the equation into two parts. The real parameter that is inserted into the KdVB equation enables us to play with the splitted parts. The real parameter enables to write the each splitted equation as close to the Korteweg de Vries (KdV) equation as we wish and as far from the Burgers equation as we wish or vice a versa. Then we solve the splitted parts numerically and compose the solutions to obtained the integrator for the KdVB equation. Finally we present some numerical experiments for the solution of the KdV, Burger’s and KdVB equations. The numerical experiments shows that the new splitting gives feasible and valid results.Article Operator Splitting of the Kdv-Burgers Type Equation With Fast and Slow Dynamics(2015) Aydın, Ayhan; Karasözen, BülentThe Korteweg de Vries-Burgers (KdV-Burgers) type equation arising from the discretiza tion of the viscous Burgers equation with fast dispersion and slow diffusion is solved using operator splitting. The dispersive and diffusive parts are discretized in space by second order conservative finite differences. The resulting system of ordinary differential equations are composed using the time reversible Strang splitting. The numerical results reveal that the periodicity of the solutions and the invariants of the KdV-Burgers equation are well preserved.Article Conservative Finite Difference Schemes for the Chiral Nonlinear Schrödinger Equation(Boundary Value Problems, 2015) Ismaıl, Mohammed S.; Al-basyounı, Khalil S.; Aydın, AyhanIn this paper, we derive three finite difference schemes for the chiral nonlinear Schrödinger equation (CNLS). The CNLS equation has two kinds of progressive wave solutions: bright and dark soliton. The proposed methods are implicit, unconditionally stable and of second order in space and time directions. The exact solutions and the conserved quantities are used to assess the efficiency of these methods. Numerical simulations of single bright and dark solitons are given. The interactions of two bright solitons are also displayed.Article Citation - WoS: 1Citation - Scopus: 2A Convergent Two-Level Linear Scheme for the Generalized Rosenau–kdv–rlw Equation(Tubitak Scientific & Technological Research Council Turkey, 2019) Aydın, AyhanA new convergent two-level finite difference scheme is proposed for the numerical solution of initial valueproblem of the generalized Rosenau–KdV–RLW equation. The new scheme is second-order, linear, conservative, andunconditionally stable. It contains one free parameter. The impact of the parameter to error of the numerical solutionis studied. The prior estimate of the finite difference solution is obtained. The existence, uniqueness, and convergence ofthe scheme are proved by the discrete energy method. Accuracy and reliability of the scheme are tested by simulating thesolitary wave graph of the equation. Wave generation subject to initial Gaussian condition has been studied numerically.Different wave generations are observed depending on the dispersion coefficients and the nonlinear advection term.Numerical experiments indicate that the present scheme is conservative, efficient, and of high accuracy, and well simulatesthe solitary waves for a long time.
