A multi-point numerical integrator with trigonometric coefficients for initial value problems with periodic solutions

Based on a collocation technique, we introduce a unifying approach for deriving a family of multi-point numerical integrators with trigonometric coefficients for the numerical solution of periodic initial value problems. A practical $3$-point numerical integrator is presented, whose coefficients are generalizations of classical linear multistep methods such that the coefficients are functions of an estimate of the angular frequency $\omega$. The collocation technique yields a continuous method, from which the main and complementary methods are recovered and expressed as a block matrix finite difference formula which integrates a second order differential equation over non-overlapping intervals without predictors. Some properties of the numerical integrator are investigated and presented. Numerical examples are given to illustrate the accuracy of the method.