Symmetric embedded predictor3–Corrector (EP3CM) methods with vanished phase–lag and its derivatives

Abstract

The embedded predictor–predictor–predictor–corrector finite difference pairs with three–stages of prediction and with eliminated phase-lag and its derivatives are described in this paper. We use the symbol (EP3CM) because the new proposed methods are of three stages of prediction. It is noted that the first stage of the predictor of the new proposed finite difference pairs is based on the linear ten–step symmetric method of Quinlan–Tremaine [1]. The new finite difference pair is called non–linear or hybrid or Runke–Kutta type because has four–stages. The methods proposed in this paper can be used for the numerical solution of: (1) initial–value problems (IVPs) with oscillatory and/or periodical solutions, (2) boundary–value problems (IVPs) with oscillatory and/or periodical solutions, (3) orbital problems and (4) the Schrödinger equation and related problems.The methods proposed in this paper belong to the embedded finite difference pairs. The numerical and theoretical results show the efficiency of the new proposed embedded methods.