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.