Piecewise-linearized methods for initial-value problems with oscillating solutions

Two piecewise-linearized methods for the numerical solution of nonlinear second-order differential equations which contain the first-order derivative are presented. These methods provide piecewise analytical solutions in open intervals, result in explicit finite difference equations for both the displacement and the velocity or only the displacement, at nodal points, and continuous solutions everywhere. One of the piecewise-linearized methods is a two-level method that provides smooth solutions and is self-starting; the other is a non-self-starting three-level technique. The paper also presents several iterative schemes, a time-linearized three-point technique and a time-linearized Numerov method. An extensive assessment of the piecewise-linearized methods is carried out in order to assess their accuracy in nine examples, and it is shown that, in some cases, these techniques provide as accurate results as exponentially- or trigonometrically-fitting methods that do require an estimate of the dominant frequency of the solution. The piecewise-linearized methods presented here do not require such an estimate at all, but may exhibit spurious fixed points/attractors in nonlinear problems and their accuracy is a strong function of both the nonlinearities and the time step. However, they are applicable to nonlinear second-order differential equations which contain the first-order derivative.