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A Runge-Kutta method is developed for the numerical solution of initial-value problems with oscillating solution. Based on the Runge-Kutta Fehlberg 2(3) method, a Runge-Kutta method with phase-lag of order infinity is developed. Based on these methods we produce a new embedded Runge-Kutta Fehlberg 2(3) method with phase-lag of order infinity. This method is called as Runge-Kutta Fehlberg Phase Fitted method (RKFPF). The numerical results indicate that this new method is much more efficient, compared with other well-known Runge-Kutta methods, for the numerical solution of differential equations with oscillating solution, using variable step size.

The Fehlberg (4, 5) formulas appear to be the most effective Runge–Kutta formulas of moderate order for the solution of non-stiff ordinary differential equations. A simple; cheap scheme is developed for codes based on these formulas which detects stiffness with considerable reliability. If a user accidentally or unknowingly tries to integrate a stiff problem, a code with this scheme can warn him that it will be extremely inefficient and that he should switch to a code aimed specifically at such problems.