Two new embedded pairs of explicit Runge-Kutta methods adapted to the numerical solution of oscillatory problems

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.