Statistics for In this paper numerical methods involving higher order derivatives for the solution of periodic initial value problems of second order differential equations are derived. The methods depend upon a parameter p > 0 and reduce to their classical counter parts as p ? 0. The methods are periodically stable when the parameter p is chosen as the square of the frequency of the linear homogeneous equation. The numerical methods involving derivatives of order up to 2q are of polynomial order 2q and trigonometric order one. Numerical results are presented for both the linear and nonlinear problems. The applicability of implicit adaptive methods to linear systems is illustrated. © 1982.