A class of two-step explicit methods for periodic IVPs

In this paper, we present a class of non-linear explicit second-order methods for solving one-dimensional periodic initial value problems (IVPs), which are P-stable in the sense of Lambert and Watson and of high-order phase-lag error. For multi-dimensional problems, we introduce a special vector operation such that our methods can be extended to be vector-applicable directly. However, it is not sufficient to investigate the stability behavior of the vector form of the methods based on the scalar test equation y″ = −λ2y. After introducing a new test system and extending the definitions of the interval of periodicity and the phase-lag order, we discuss the stability property and the phase-lag of the vector-applicable methods. The methods are P-stable and of high phase-lag for one-frequency problems. For multi-frequency problems, methods are of only second phase-lag order and may have not a primary interval of periodicity. Some numerical results illustrate the conclusions.