Error bounds for explicit ERKN integrators for systems of multi-frequency oscillatory second-order differential equations

A substantial issue of numerical analysis is concerned with the investigation and estimation of the errors. In this paper, we pay attention to the error analysis for the extended Runge–Kutta–Nyström (ERKN) integrators proposed by Wu et al. (2010) [30] for systems of multi-frequency oscillatory second-order differential equations q″(t)+Mq(t)=f(q(t))q″(t)+Mq(t)=f(q(t)). The ERKN integrators are important generalizations of the classical Runge–Kutta–Nyström methods in the sense that both the updates and internal stages have been reformed so that the quantitative behavior of ERKN integrators is adapted to the oscillatory properties of the true solution. By the expansions for the errors of explicit ERKN integrators, we derive stiff order conditions up to order three and present the error bounds.We show that the explicit ERKN integrator fulfilling stiff order p converges with order p, and for an important particular case where M is a symmetric and positive semi-definite matrix, the error bound of  ‖qn−q(tn)‖‖qn−q(tn)‖is independent of  ‖M‖‖M‖ (‖⋅‖‖⋅‖denotes the Euclidean norm). The stiff order conditions provided in the error analysis allow us to design new and efficient explicit ERKN integrators for multi-frequency oscillatory systems. We propose a novel explicit third order multi-frequency and multidimensional ERKN integrator with minimal dispersion error and dissipation error. Numerical experiments carried out show that our new explicit multi-frequency and multidimensional ERKN integrator is more efficient than various other existing effective methods in the scientific literature. We use the first problem to show that the methods perform well with nonsymmetric matrices. In particular, for the well-known Fermi–Pasta–Ulam problem, the numerical behavior of our new explicit ERKN integrator supports our theoretical analysis.