Optimized Runge–Kutta pairs for problems with oscillating solutions

Three types of methods for integrating periodic initial value problems are presented. These methods are (i) phase-fitted, (ii) zero dissipation (iii) both zero dissipative and phase fitted. Some particular modifications of well-known explicit Runge–Kutta pairs of orders five and four are constructed. Numerical experiments show the efficiency of the new pairs in a wide range of oscillatory problems.