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

In this paper we describe the embedded predictor–predictor–corrector methods with two–stages of prediction and with vanished phase-lag and its derivatives. Since the methods described in this paper have two–stages of prediction, the symbol (EPPCM) is used. The first stage of the predictor of the new presented algorithm is based on the linear ten–step symmetric method of Quinlan–Tremaine [1]. We call the new scheme non–linear since has three–stages. We use the methods presented in this paper on the numerical 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. We note here, that the algorithms presented in this paper belong to the embedded methods. The numerical and theoretical results show the effectiveness of the new obtained embedded finite difference pairs.