Exponentially fitted explicit Runge–Kutta–Nyström methods

Abstract

Exponentially fitted Runge–Kutta–Nyström (EFRKN) methods for the numerical integration of second-order IVPs with oscillatory solutions are derived. These methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp( λt),exp(− λt)}, λ∈ C , or equivalently {sin( ωt),cos( ωt)} when λ=i ω, ω∈ R . Explicit EFRKN methods with two and three stages and algebraic orders 3 and 4 are constructed. In addition, a 4(3) embedded pair of explicit EFRKN methods based on the FSAL technique is obtained, which permits to introduce an error and step length control with a small cost added. Some numerical experiments show the efficiency of our explicit EFRKN methods when they are compared with other exponential fitting type codes proposed in the scientific literature.