New explicit adapted Numerov methods for second-order oscillatory differential equations

This paper is concerned with a novel class of explicit adapted Numerov (AN) methods for solving initial value problems of second-order oscillatory differential equations of the form y″+ω2y=f(x,y). The scheme of the methods incorporate the special structure of the problem carried by the term ω2y, so that they can integrate the linear equation y″+ω2y=0 without truncation error. A new type of simplifying assumption is proposed on the linear operators related to the quadratures that lead to the AN scheme. A practical AN method of order five and three AN methods of order six are derived. The phase and stability properties of the new methods are analyzed. The results of numerical experiments show the robustness and competence of the new methods compared with some highly efficient codes in the recent literature.