Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions

We construct explicit Runge–Kutta (–Nystrom) methods for the integration of first (and second) order differential equations having an oscillatory solution. Special attention is paid to the phase errors (or dispersion) of the dominant components in the numerical oscillations when these methods are applied to a linear, homogeneous test model. RK(N) methods are constructed which are dispersive of orders up to 10, whereas the (algebraic) order of accuracy is only 2 or 3. Application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced.