Symmetric embedded predictor–corrector (EP 2 CM) methods with vanished phase–lag and its derivatives for second order problems

Embedded predictor–corrector methods with two–stages of prediction and with vanished phase-lag and its derivatives are described in this paper. We give the symbol (EP 2 CM) since these methods have two–stages of prediction. The first stage of the predictor of the new algorithm is based on the linear eight–step symmetric method of Quinlan–Tremaine [1]. The new scheme is non–linear since has three–stages. These methods can be used on the approximate solution of: 1. initial–value problems (IVPs) with oscillatory solutions, 2. boundary–value problems (IVPs) with oscillatory solutions, 3. orbital problems 4. the Schrödinger equation and related problems. The new presented algorithm belongs to the embedded methods. The numerical and theoretical achievements show the efficiency of the new produced embedded scheme.