A novel family of P-stable symmetric extended linear multistep methods for oscillators

•Scheme of Phase-fitted and P-stable symmetric extended linear multistep (SELM) methods.•Order conditions for modified linear multistep methods.•Practical explicit and implicit SELM methods.•Eight-step SELM predictor-corrector method of order ten.•Numerical experiments showing the effectiveness of the SELM methods.

A novel family of phase-fitted and P-stable symmetric extended linear multistep (SELM) methods for solving initial value problems of second-order oscillatory differential equations are developed. SELM methods are naturally P-stable. The order conditions are presented. Based on the harmonic fitting condition and order conditions, new explicit SELM methods of order two, order four, order six and order eight, respectively and implicit SELM methods of order four, order six, order eight and order ten, respectively, are derived and the local error for each method is obtained. A new eight-step SELM predictor-corrector pair of order ten is obtained. Numerical experiments are accompanied to show the high accuracy and competence of the new SELM methods compared with some highly efficient symmetric multistep methods in the literature.