Embedded Runge-Kutta methods for periodic initial-value problems

Some embedded Runge-Kutta methods with minimal phase-lag for second-order periodic initial-value problems are developed. It should be noted that these embedded methods are based on the Runge-Kutta methods of algebraic order three, and on a new error estimation introduced in this paper. The numerical results indicate that these new methods are efficient for the numerical solution of differential equations with periodical solution, using variable step-size.