High order adaptive methods of Nystrِm-Cowell type

A class of unconditionally stable multistep methods is discussed for solving initial-value problems of second-order differential equations which have periodic or quasiperiodic solutions. This situation frequently occurs in celestial mechanics, in nonlinear oscillations and various other situations. The methods depend upon a parameter ω > 0, and integrate exactly trigonometric functions along with algebraic polynomials. In this paper we show a procedure for the construction of adaptive Nyström-Cowell formulas of arbitrarily high order of accuracy, and reduce to the classical Nyström-Cowell methods for ω = 0. Our methods compare advantageously with other methods.