Runge-Kutta methods for Hamiltonian systems in non-standard symplectic two-form

Runge-Kutta methods are applied to Hamiltonian systems on Poisson manifolds with a nonstandard symplectic two-form. It has been shown that the Gauss Legendre Runge-Kutta (GLRK) methods and combination of the partitioned Runge-Rutta methods of Lobatto IIIA and IIIb type are symplectic up to the second order in terms of the step size. Numerical results on Lotka-Volterra and Kermack-McKendrick epidemic disease model reveals that the application of the symplectic Runge-Kutta methods preserves the integral invariants of the underlying system for long-time computations.


Poisson integrators for Volterra lattice equations
Ergenc, T; Karasözen, Bülent (2006-06-01)
The Volterra lattice equations are completely integrable and possess bi-Hamiltonian structure. They are integrated using partitioned Lobatto IIIA-B methods which preserve the Poisson structure. Modified equations are derived for the symplectic Euler and second order Lobatto IIIA-B method. Numerical results confirm preservation of the corresponding Hamiltonians, Casimirs, quadratic and cubic integrals in the long-term with different orders of accuracy. (c) 2005 IMACS. Published by Elsevier B.V. All rights re...
Runge-Kutta scheme for stochastic optimal control problems
Öz Bakan, Hacer; Weber, Gerhard Wilhelm; Yılmaz, Fikriye; Department of Financial Mathematics (2017)
In this thesis, we analyze Runge-Kutta scheme for the numerical solutions of stochastic optimal control problems by using discretize-then-optimize approach. Firstly, we dis- cretize the cost functional and the state equation with the help of Runge-Kutta schemes. Then, we state the discrete Lagrangian and take the partial derivative of it with respect to its variables to get the discrete optimality system. By comparing the continuous and discrete optimality conditions, we find a relationship between the Runge...
Symplectic and multisymplectic Lobatto methods for the "good" Boussinesq equation
AYDIN, AYHAN; Karasözen, Bülent (2008-08-01)
In this paper, we construct second order symplectic and multisymplectic integrators for the "good" Boussineq equation using the two-stage Lobatto IIIA-IIIB partitioned Runge-Kutta method, which yield an explicit scheme and is equivalent to the classical central difference approximation to the second order spatial derivative. Numerical dispersion properties and the stability of both integrators are investigated. Numerical results for different solitary wave solutions confirm the excellent long time behavior ...
DRBEM applications in fluid dynamics problems and DQM solutions of hyperbolic equations
Pekmen, Bengisen; Tezer Sezgin, Münevver; Department of Scientific Computing (2014)
In this thesis, problems of fluid dynamics defined by the two-dimensional convection-diffusion type partial differential equations (PDEs) are solved using the dual reciprocity boundary element method (DRBEM). The terms other than the Laplacian are treated as inhomogeneous terms in the DRBEM application. Once the both sides are multiplied by the fundamental solution of Laplace equation, and then integrated over the domain, all the domain integrals are transformed to boundary integrals using the Green's ident...
MUSTAFA, O; Sever, Ramazan (1991-10-01)
The shifted 1/N expansion method has been extended to solve the Klein-Gordon equation with both scalar and vector potentials. The calculations are carried out to the third-order correction in the energy series. The analytical results are applied to a linear scalar potential to obtain the relativistic energy eigenvalues. Our numerical results are compared with those obtained by Gunion and Li [Phys. Rev. D 12, 3583 (1975)].
Citation Formats
B. Karasözen, “Runge-Kutta methods for Hamiltonian systems in non-standard symplectic two-form,” INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS, pp. 113–122, 1998, Accessed: 00, 2020. [Online]. Available: