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Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials. (English) Zbl 0927.65097
The paper is concerned with initial value problems for ordinary differential equations (ODEs) whose solutions are a priori known to oscillate with a known frequency. Runge-Kutta methods are applied to first-order problems and Runge-Kutta-Nyström methods to second-order problems. P. Albrecht’s approach [SIAM J. Numer. Anal. 24, 391-406 (1987; Zbl 0617.65067); Teubner-Texte Math. 104, 8-18 (1988; Zbl 0682.65041)] is considered and the concept of trigonometric order of the method is used. The Runge-Kutta and Runge-Kutta-Nyström methods derived in this paper have low orders (1 or 2) and integrate trigonometric polynomials exactly. Two numerical examples are performed for comparison to others-Runge-Kutta-Nyström methods.
MSC:
65L06Multistep, Runge-Kutta, and extrapolation methods
65L05Initial value problems for ODE (numerical methods)
34C25Periodic solutions of ODE
34A34Nonlinear ODE and systems, general