Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations

In the present work, a kind of trigonometric collocation methods based on Lagrange basis polynomials is developed for effectively solving multi-frequency oscillatory second-order differential equations q″(t)+Mq(t)=f(q(t))q″(t)+Mq(t)=f(q(t)). The properties of the obtained methods are investigated. It is shown that the convergent condition of these methods is independent of ‖M‖‖M‖, which is very crucial for solving oscillatory systems. A fourth-order scheme of the methods is presented. Numerical experiments are implemented to show the remarkable efficiency of the methods proposed in this paper.