A new kind of high-efficient and high-accurate P-stable Obrechkoff three-step method for periodic initial-value problems

In order to improve the efficiency and accuracy of the previous Obrechkoff method, in this paper we put forward a new kind of P-stable three-step Obrechkoff method of O(h¹⁰) for periodic initial-value problems. By using a new structure and an embedded high accurate first-order derivative formula, we can avoid time-consuming iterative calculation to obtain the high-order derivatives. By taking advantage of new trigonometrically-fitting scheme we can make both the main structure and the first-order derivative formula to be P-stable. We apply our new method to three periodic problems and compare it with the previous three Obrechkoff methods. Numerical results demonstrate that our new method is superior over the previous ones in accuracy, efficiency and stability.