A new high efficient and high accurate Obrechkoff four-step method for the periodic nonlinear undamped Duffing's equation
Based on the idea of the previous Obrechkoff's 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, H 0 2 ? 16.33 . The numerical test to the well known problem, the nonlinear undamped Duffing's 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 Obrechkoff's two-step method. The ultimate accuracy of the new method can reach about 510?13 5  10 ?13 , 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 (10?11) ( 10 ?11 ) , to below 10?14.
high-order derivative multistep method obrechkoff method second-order initial value problem with periodic solutions numerical solution to the duffing equation