Multi-step Nyström methods for general second-order initial value problems y″(t) = f(t,y(t), y′(t))

Abstract

ABSTRACTA class of multi-step Nyström methods for solving numerically general second-order initial value problems are proposed and developed. The new methods inherit the main frameworks of classical Nyström methods and make use of the information of previous steps. Order conditions are deduced by using the theory of Nyström-series based on the set of Nyström-trees, and two explicit methods with convergence orders three and four, respectively, are constructed. The linear stability of the new methods is analysed. Numerical results show that our new methods are more efficient in comparison with Runge–Kutta–Nyström methods and other well-known high quality methods proposed in the scientific literature.