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 ω 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)}, the estimate of the local error behaves as O(h⁴) and the high-order formula has fourth-order accuracy when the stepsize h→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.