Abstract
The existence and construction of symplectic 2s-stage variable coefficients Runge-Kutta (RK) methods that integrate exactly IVPs whose solution is a trigonometrical polynomial of order s with a given frequency ω is considered. The resulting methods, that can be considered as trigonometrical collocation methods, are fully implicit, symmetric and symplectic RK methods with variable nodes and coefficients that are even functions of ν=ω h (h is the step size), and for ω→0 they tend to the conventional RK Gauss methods. The present analysis extends previous results on two-stage symplectic exponentially fitted integrators of Van de Vyver (Comput. Phys. Commun. 174: 255–262, 2006) and Calvo et al. (J. Comput. Appl. Math. 218: 421–434, 2008) to symmetric and symplectic trigonometrically fitted methods of high order. The algebraic order of the trigonometrically fitted symmetric and symplectic 2s-stage methods is shown to be 4s like in conventional RK Gauss methods. Finally, some numerical experiments with oscillatory Hamiltonian systems are presented.
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Communicated by Timo Eirola.
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Calvo, M., Franco, J.M., Montijano, J.I. et al. On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages. Bit Numer Math 50, 3–21 (2010). https://doi.org/10.1007/s10543-010-0250-z
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DOI: https://doi.org/10.1007/s10543-010-0250-z