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In this paper we develop a family of three 8-step methods, optimized for thenumerical integration of oscillatory ordinary differential equations. We havenullified the phase-lag of the methods and the first r derivatives, wherer=1,2,3. We show that with this new technique, the method gains efficiency witheach derivative of the phase-lag nullified. This is the case for theintegration of both the Schrodinger equation and the N-body problem. A localtruncation error analysis is performed, which, for the case of the Schrodingerequation, also shows the connection of the error and the energy, revealing theimportance of the zero phase-lag derivatives. Also the stability analysis showsthat the methods with more derivatives vanished, have a bigger interval ofperiodicity.

A Family of Multistep Methods with Zero Phase-Lag and Derivatives for the Numerical Integration of Oscillatory ODEs