A variable step-size implementation of the hybrid Nyström method for integrating Hamiltonian and stiff differential systems

The approximate solution to second-order Hamiltonian and stiff differential systems is obtained using an efficient hybrid Nyström method (HNM) in this manuscript. The development of the method considers three hybrid points that are selected by optimizing the local truncation errors of the main formulas. The properties of the proposed HNM are studied. An embedding-like procedure is explored and run in variable step-size mode to improve the accuracy of the HNM. The numerical integration of some second-order Hamiltonian and stiff model problems, such as the well-known Vander Pol, Fermi-Pasta-Ulam, and Duffing problems, demonstrate the improved impact of our devised error estimation and control strategy. Finally, it is essential to note that the proposed technique is efficient in terms of computational cost and maximum global errors.