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This paper focuses on the study of explicit two-step Birkhoff-Hermite (TSBH) integrators for solving nonlinear highly oscillatory second-order ordinary differential equations (ODEs). These new TSBH integrators, involving  r th-order derivatives, are of order 2 r  + 2. Nonlinear stability and convergence of the integrators are analyzed. Two practical TSBH integrators of order four and order six are derived. Numerical results demonstrate the advantage and efficiency of the proposed integrators in comparison with the existing numerical methods reported in the literature.

A novel class of explicit two-step Birkhoff-Hermite integrators for highly oscillatory second-order differential equations