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.