Two new embedded pairs of explicit Runge–Kutta methods adapted to the numerical solution of oscillatory problems J.M. Franco ⇑ , Y. Khiar, L. Rández IUMA, Universidad de Zaragoza, Departamento de Matemática Aplicada, Spain article info Keywords: Adapted Runge–Kutta methods Embedded pairs Oscillatory problems abstract The construction of new embedded pairs of explicit Runge–Kutta methods specially adapted to the numerical solution of oscillatory problems is analyzed. Based on the order conditions for this class of methods, two new embedded pairs of orders 4(3) and 6(4) which require five and seven stages per step, respectively, are constructed. The derivation of the new embedded pairs is carried out paying special attention to the minimization of the principal term of the local truncation error as well as the dispersion and dissipation errors of the higher order formula. Several numerical experiments are carried out to show the efficiency of the new embedded pairs when they are compared with some standard and specially adapted pairs proposed in the scientific literature for solving oscillatory problems. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction In this paper we analyze the construction of embedded pairs of explicit Runge–Kutta methods (RK methods) specially adapted to the numerical solution of oscillatory differential systems. These oscillatory problems often arise in different fields of applied sciences such as celestial mechanics, astrophysics, chemistry, electronics, molecular dynamics and engineering (see [1–3]), and they can be modeled by initial value problems (IVPs) of the form y 0 ðtÞ¼ f ðt; yðtÞÞ; yðt 0 Þ¼ y 0 2 R m ; t 2½t 0 ; T ; ð1Þ where for simplicity f ðt; yÞ is assumed to be sufficiently smooth, so that the IVP (1) has a unique solution. Since the analytical solutions of these IVPs are usually not available, they can be solved by using general purpose numerical methods or by using codes specially adapted to the oscillatory behavior of their solutions. The design and construction of RK methods specially adapted to the numerical solution of oscillatory IVPs (1) has been considered by several authors (see [4–19] and references therein). The coefficients of these methods usually depend on the parameter m ¼ x h, where h is the integration step-size and x represents an approximation of the main frequency of the problem. The aim of these methods is to use the particular structure of the differential system (1) and the form of its solution to derive more accurate and efficient algorithms than the general purpose RK algorithms for such a type of problems. We mention the pioneer paper of Bettis [4], in which adapted explicit RK algorithms with 3 and 4 stages were introduced for solving differential systems with oscillatory solutions. Subsequently, Franco [10] analyzed the necessary and sufficient order conditions for this class of RK methods and he constructed explicit RK algorithms (up to order 4) as well as embedded pairs of explicit RK methods with orders 4 and 3. More recently, some authors [13,14,18] have proposed adapted versions of the http://dx.doi.org/10.1016/j.amc.2014.11.097 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: jmfranco@unizar.es (J.M. Franco), yaskhiarviana@gmail.com (Y. Khiar), randez@unizar.es (L. Rández). Applied Mathematics and Computation 252 (2015) 45–57 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc