SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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An energy-preserving exponentially-fitted continuous stage Runge-Kutta method for Hamiltonian systems

Y. Miyatake and T. Matsuo

Abstract
Recently, the construction of symplectic exponentially-fitted Runge-Kutta (EFRK) methods for the numerical integration of Hamiltonian systems with periodic or oscillatory solutions have been attracting a lot of interest (see [1, 3], for example). In this talk, from the standpoint of geometric integration, we consider a derivation of energy-preserving exponentially-fitted methods. For this aim, we show sufficient conditions for energy-preservation in terms of the coefficients of continuous stage RK methods (continuous stage RK methods were introduced by Hairer [2]), and extend the theory of EFRK methods to the context of continuous stage RK methods. In this talk, by combining these two theories, we derive second and fourth order energy-preserving exponentially-fitted schemes.

Bibliography
[1] M. Calvo, J. M. Franco, J. I. Montijano and L. Rández, Structure preservation of exponentially fitted Runge-Kutta methods, J. Comput. Appl. Math., 218 (2008) 421-434.
[2] E. Hairer, Energy-preserving variant of collocation methods, JNAIAM J. Numer. Anal. Ind. Appl. Math., 5 (2010) 73-84.
[3] G. Vanden Berghe and M. Van Daele, Symplectic exponentially-fitted four-stage Runge-Kutta methods of the Gauss type, Numer. Algor., 56 (2011) 591-608.

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