Explicit two-step high-accuracy hybrid methods with minimal phase-lag for y″ = f(x, y) and their application to the one-dimensional Schrödinger equation

In this paper, two families of explicit two-step sixth and eighth algebraic order hybrid methods with minimal phaselag are developed for the numerical integration of special second-order periodic initial-value problems. These methods have the advantage of higher algebraic accuracy and minimal phase-lag compared with some methods in [1, 2, 4–8, 11–14]. The methods proposed in this paper may be considered as a generalization of some methods in [1, 5, 7, 8, 12]. An application to the one-dimensional Schrodinger equation on the resonance problem indicates that these new methods are generally more accurate than some methods in [5–8, 2, 11, 13, 14].