A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped Duffingʹs equation Original Research Article

Based on the idea of the previous Obrechkoffʹs two-step method, a new kind of four-step numerical method with free parameters is developed for the second order initial-value problems with oscillation solutions. By using high-order derivatives and apropos first-order derivative formula, the new method has greatly improved the accuracy of the numerical solution. Although this is a multistep method, it still has a remarkably wide interval of periodicity, image. The numerical test to the well known problem, the nonlinear undamped Duffingʹs equation forced by a harmonic function, shows that the new method gives the solution with four to five orders higher than those by the previous Obrechkoffʹs two-step method. The ultimate accuracy of the new method can reach about image, which is much better than the one the previous method could. Furthermore, the new method shows the great superiority in efficiency due to a reasonable arrangement of the structure. To finish the same computational task, the new method can take only about 20% CPU time consumed by the previous method. By using the new method, one can find a better ‘exact’ solution to this problem, reducing the error tolerance of the one widely used method image, to below 10−14.