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 ω. 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.