Proficient trigonometrical-fitted two-derivative multistep collocation methods in predictor-corrector approach: Application to perturbed Kepler problem.

An efficient trigonometrical-fitted two-derivative multistep collocation (TF-TDMC) method using Legendre polynomials up to order five as the basis functions, has been developed for solving second-order ordinary differential equations with oscillatory solution effectively. Interpolation method of approximated power series and collocation technique of its second and third derivative are implemented in the construction of the methods. Two-derivative multistep collocation methods are developed in predictor and corrector form with varying collocation and interpolation points. Later, trigonometrically-fitting technique is implemented into TF-TDMC method, using the linear combination of trigonometrical functions, to produce frequency-dependent coefficients in TF-TDMC method. The stability of the TF-TDMC method, with fitted parameters, is thoroughly analyzed and has been proven to achieve zero stability. Stability polynomials and regions for predictor and corrector of TF-TDMC method are developed and plotted. In the operation of the TF-TDMC method, initial conditions and the frequency for each problem (based on the exact solutions) are identified. The frequency-dependent coefficients are then adjusted according to the identified frequency. Predictor and corrector steps are implemented to estimate and refine the values of the dependent variable and its derivative, ensuring that convergence is achieved. A numerical experiment demonstrates that the proposed method significantly outperforms other existing methods in the literature, achieving the lowest maximum global error with moderate computational time across all step sizes for solving second-order ordinary differential equations with oscillatory solutions. Additionally, it effectively addresses real-world perturbed Kepler problems. The results include a detailed discussion and analysis of the numerical performance.•An efficient two-derivative multistep collocation method in predictor-corrector mode with trigonometrically-fitting technique (TF-TDMC) is developed for direct solving second-order ordinary differential equations with oscillatory solution.•TF-TDMC method has been proved to acquire zero-stability and its stability region is analyzed.•TF-TDMC method is the best among all selected methods in solving second-order ordinary differential equations with oscillatory solution, including perturbed Kepler problem.