High algebraic order Runge-Kutta type two-step method with vanished phase-lag and its first, second, third, fourth, fifth and sixth derivatives

A Runge-Kutta type twelfth algebraic order two-step method with vanished phase-lag and its first, second, third, fourth, fifth and sixth derivatives are developed in this paper. The construction of the method, the local truncation error (lte) of the newly obtained method, the comparative error analysis of the new method with the corresponding method with constant coefficients and the stability (interval of periodicity) of the new method using frequency for the scalar test equation different than the frequency used in the scalar test equation for phase-lag analysis are studied in this paper. Finally, an application of the newly obtained method to the coupled differential equations of the Schrödinger type is also presented in this paper. From the presented numerical results, the efficiency of the newly proposed method is shown. It is noted that for the first time in the literature a multistep method with vanished phase-lag and its derivatives up to sixth order is developed.