New embedded explicit pairs of exponentially fitted Runge–Kutta methods

Two new embedded pairs of exponentially fitted explicit Runge–Kutta methods with four and five stages for the numerical integration of initial value problems with oscillatory or periodic solutions are developed. In these methods, for a given fixed ωω the coefficients of the formulae of the pair are selected so that they integrate exactly systems with solutions in the linear space generated by {sinh(ωt),cosh(ωt)}{sinh(ωt),cosh(ωt)}, the estimate of the local error behaves as O(h4)O(h4) and the high-order formula has fourth-order accuracy when the stepsize h→0h→0. These new pairs are compared with another one proposed by Franco [J.M. Franco, An embedded pair of exponentially fitted explicit Runge–Kutta methods, J. Comput. Appl. Math. 149 (2002) 407–414] on several problems to test the efficiency of the new methods.