Importance of the first-order derivative formula in the Obrechkoff method

In this paper we present a delicately designed numerical experiment to explore the relationship between the accuracy of the first-order derivative (FOD) formula and the one of the main structure in an Obrechkoff method. We choose three two-step P-stable Obrechkoff methods as the main structure, which are available from the previous published literature, their local truncation error (LTE(h)) ranging from O(h8yn(8)) to O(h12yn(12)), and six FOD formulas, of which the former five ones have the similar structures and the sixth is the 'exact' value of the FOD, their LTE(h) arranged from O(h4yn(5)) to O(h13yn(14)) (we will use O(yn(m)) to represent the order of a LTE(h)), as the main ingredients for our numerical experiment. We survey the numerical results by integrating the Duffing equation without damping and compare them with the 'exact' solution, and find out how its numerical accuracy is affected by a FOD formula. The experiment shows that a high accurate FOD formula can greatly improve the numerical accuracy of an Obrechkoff method for a given main structure, and the error in the numerical solution decreases with the order of the LTE(h) of a FOD formula, only when the order of LTE(h) of the FOD formula is equal to or higher than the one of the main structure, the accuracy of the Obrechkoff method is no longer affected by the approximation of the FOD formula.