Two-frequency trigonometrically-fitted and symmetric linear multi-step methods for second-order oscillators

This paper focuses on symmetric linear multi-step methods of Numerov-type for initial-value problems with two principal frequencies. A new explicit two-frequency trigonometrically fitted (TFTF) and symmetric two-step method of order two, and an explicit TFTF symmetric four-step method of order four are constructed. A characteristic feature of the new methods is that they can integrate without truncation error the problem whose solution is a linear combination of the harmonic oscillators with these two frequencies. The stability and phase lags of the new methods are analyzed. Numerical experiments show the high effectiveness and robustness of the new methods in comparison with some well-known one-frequency trigonometrically/exponentially fitted symmetric multi-step methods in the recent literature.