A new class of two-step P-stable TFPL methods for the numerical solution of second-order IVPs with oscillating solutions

A new class of two-step linear symmetric methods is introduced for the numerical solution of second-order initial value problems that have highly oscillatory solutions. In this class, for the first time in the literature, we calculate the coefficients of the method by a combination of a trigonometrically fitted (TF) method and a vanishing of the phase lag and some of its derivatives (VSDPL) method, and we construct a new class of methods that has the properties of TF methods and VSDPL methods, which we call the TFPL method. This method is of algebraic order 8, and has an important P-stability property. The main structure of the method is multiderivative, and the combined phases were applied to expand the stability interval and to achieve P-stability. The advantage of the method in comparison with similar methods in terms of efficiency, accuracy, and stability is shown by its implementation in some important problems, the undamped Duffing equation, etc. (C) 2018 Elsevier B.V. All rights reserved.