Adaptive two-derivative Runge–Kutta–Nyström method with trigonometric fitting approach

A fifth-order trigonometrically-fitted explicit two-derivative Runge–Kutta–Nyström method with an adaptive step size strategy, denoted as ATFRKN5, is proposed for efficiently solving second-order ordinary differential equations of the form u″(x) = f(x, u(x)) that exhibit oscillatory behavior. The order conditions of the method are derived using Taylor series expansion, enabling the construction of two-derivative Runge–Kutta–Nyström schemes up to orders four and five. Trigonometric fitting is applied by incorporating the basis functions eiλx and e−iλx, λ ∈ ℝ, allowing the method to adapt naturally to problems with dominant frequencies. A stability analysis is performed, and the corresponding stability regions are presented and discussed. An adaptive algorithm is developed to enhance robustness and accuracy across varying frequency profiles. Numerical experiments, including linear harmonic oscillators, nonlinear oscillatory models, and a stiff asymmetric Duffing oscillator, demonstrate that ATFRKN5 achieves significantly higher accuracy and improved error control while requiring fewer function evaluations compared with existing methods of the same order. The results confirm that ATFRKN5 is particularly effective for periodic and highly oscillatory problems, especially those with nonlinear or stiff characteristics.