Symmetric embedded predictor5–corrector (EP5CM) full in phase methods

An embedded predictor5–corrector scheme is presented in this paper. The meaning of the terminology predictor5 is five layers of prediction. We symbolize the new embedded algorithm as (EP5CM). For the scheme (EP5CM), the coefficients are determined in order the algorithm to have vanished the phase–lag and its derivatives of the highest possible order for each layer. The linear symmetric ten–step method of Quinlan–Tremaine [1] is used as the first layer (predictor) of the new introduced scheme. The new method can be applied on the approximate solution of: (i) special initial–value problems (IVPs) with oscillating and/or periodical solution, (ii) boundary–value problems (BVPs) with oscillating and/or periodical solution, (iii) orbital problems and (iv) the Schrödinger equation and related problems. The numerical and theoretical achievements show the efficiency of the new introduced algorithms.