A Family of Multistep Methods with Zero Phase-Lag and Derivatives for   the Numerical Integration of Oscillatory ODEs
  In this paper we develop a family of three 8-step methods, optimized for the numerical integration of oscillatory ordinary differential equations. We have nullified the phase-lag of the methods and the first r derivatives, where r=1,2,3. We show that with this new technique, the method gains efficiency with each derivative of the phase-lag nullified. This is the case for the integration of both the Schrodinger equation and the N-body problem. A local truncation error analysis is performed, which, for the case of the Schrodinger equation, also shows the connection of the error and the energy, revealing the importance of the zero phase-lag derivatives. Also the stability analysis shows that the methods with more derivatives vanished, have a bigger interval of periodicity. 
Z. A. Anastassi, D. S. Vlachos and T. E. Simos