The Use of Phase-Lag Derivatives in the Numerical Integration of ODEs with Oscillating Solutions

In this paper we consider the fitting of the coefficients of a numerical method, not only due to the nullification of the phase-lag, but also to its derivatives. We show that the method gains efficiency with each derivative of the phase-lag nullified for various problems with oscillating solutions. The analysis of the local truncation error analysis and the stability of the methods show the importance of zero phase-lag derivatives when integrating oscillatory differential equations.