Frequency determination and step-length control for exponentially-fitted Runge–Kutta methods

An exponentially fitted Runge–Kutta (EFRK) fifth-order method with six stages is constructed, which exactly integrates first-order differential initial-value problems whose solutions are linear combinations of functions of the form {exp(ωx),exp(−ωx)}, (ω∈R or iR). By combining this EFRK method with an equivalent classical embedded (4,5) Runge–Kutta method, a technique is developed for the estimation of the occurring ω-values. Error and step-length control is carried out by using the Richardson extrapolation procedure. Some numerical experiments show the efficiency of the introduced methods.