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Symplectic conditions for exponential fitting Runge-Kutta-Nyström methods. (English) Zbl 1086.65120
Summary: Symplecticity conditions easy to handle for constructing symplectic Runge-Kutta-Nyström methods fitted to trigonometric functions are given. These conditions generalize that of {\it Yu. B. Suris} [Zh. Vychisl. Mat. Mat. Fiz. 29, No. 2, 202--211 (1989; Zbl 0686.65039)] when the frequencies tends to zero.

MSC:
65P10Numerical methods for Hamiltonian systems including symplectic integrators
37M15Symplectic integrators (dynamical systems)
65L06Multistep, Runge-Kutta, and extrapolation methods
34A26Geometric methods in differential equations
34A34Nonlinear ODE and systems, general
65L05Initial value problems for ODE (numerical methods)
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Full Text: DOI
References:
[1] Suris, Y. B.: On the canonicity of mappings that can be generated by methods of Runge-Kutta type for integrating systems :$x = -\partialu/\partialx $(in russian). U.S.S.R. comput. Math. and math. Phys. 29, No. 1, 138-144 (1990)
[2] Hairer, E.; Lubich, C.; Wanner, G.: Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. (2002) · Zbl 0994.65135
[3] Arnold, V. I.: Mathematical methods of classical mechanics. (1989)
[4] Sanz-Serna, J. M.: Symplectic integrators for Hamiltonian problems: an overview. Acta numerica 1, 243-286 (1992) · Zbl 0762.65043
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[7] Calvo, M. P.; Sanz-Serna, J. M.: High-order symplectic Runge-Kutta-Nyström methods. SIAM J. Sci. comput. 14, 1237-1252 (1993) · Zbl 0787.65056
[8] Calvo, M. P.; Sanz-Serna, J. M.: The development of variable-step symplectic integrators with application to the two-body problem. SIAM J. Sci. comput. 14, 936-952 (1993) · Zbl 0785.65083
[9] Vigo-Aguiar, J.; Ferrándiz, J. M.: A general procedure por the adaptation of multistep algorithms to the integration of oscillatory problems. Sinum 35, No. 4, 1684-1708 (1998) · Zbl 0916.65081
[10] Vigo-Aguiar, J.; Simos, T. E.: An exponentially fitted and trigonometrically fitted method for the numerical solution of orbital problems. The astronomical journal 122, No. 3, 1656-1660 (2001)
[11] Berghe, G. Vanden; De Meyer, H.; Van Daele, M.; Van Hecke, T.: Exponentially fitted Runge-Kutta methods. Journal of computational and applied mathematics 125, No. 1--2, 107-115 (2000) · Zbl 0999.65065
[12] Simos, T. E.: An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Computer physics communications 115, No. 1, 1 (1998) · Zbl 1001.65080
[13] Simos, T. E.; Vigo-Aguiar, J.: J. exponentially fitted symplectic integrator. Phys. rev. E 67, No. 1, 016701-016707 (2003)
[14] Vigo-Aguiar, J.; Simos, T. E.; Tocino, A.: An adapted symplectic integrator for Hamiltonian problems. J. modern phys. C 12, No. 2, 225-234 (2001)