A four-step trigonometric fitted P-stable Obrechkoff method for periodic initial-value problems

In this paper, we present a new P-stable Obrechkoff four-step method, which greatly improves the performance of our previous Obrechkoff four-step method and extends its application range. By trigonometric fitting, we extend the interval of periodicity of the previous four-step method from about H 2 ∼ 16 to infinity and at the same time, we keep all its advantage in the accuracy and efficiency. We have tested the new method by four well-known problems, (1) the test-equation; (2) Stiefel and Bettis problem; (3) Duffing equation without damping; and (4) Bessel equation. The numerical results show that the new method is more accurate than any previous method. It also has great advantage in stability and efficiency.