A high-accurate and efficient Obrechkoff five-step method for undamped Duffing's equation

摘要

In this paper, we present a five-step Obrechkoff method to improve the previous two-step one for a second-order initial-value problem with the oscillatory solution. We use a special structure to construct the iterative formula, in which the higher-even-order derivatives are placed at central four nodes, and show there existence of periodic solutions in it with a remarkably wide interval of periodicity, H-0(2) similar to 16.28. By using a proper first-order derivative (1701)) formula to make this five-step method to have two advantages (a) a very high accuracy since the local truncation error (LTE) of both the main structure and the FOD formula are the same as O(h(14)); (b) a high efficiency because it avoids solving a polynomial equation with degree-nine by Picard iterative. By applying the new method to the well-known problem, the nonlinear Duffing's equation without damping, we can show that our numerical solution is four to five orders higher than the one by the previous Obrechkoff two-step method and it takes only 25% of CPU time required by the previous method to fulfil the same task. By using the new method, a better "exact" solution is found by fitting, whose error tolerance is below 5 X 10(-15), than the one widely used in the lectures, whose error tolerance is below 10(-11).