A HIGH-ACCURATE AND EFFICIENT OBRECHKOFF FIVE-STEP METHOD FOR UNDAMPED DUFFING'S EQUATION

Deyin Zhao, Zhongcheng Wang, Yongming Dai and Yuan Wang
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Deyin Zhao: Department of Physics, Shanghai University, 99 ShangDa Road, Shanghai 200444, P. R. China
Zhongcheng Wang: Department of Physics, Shanghai University, 99 ShangDa Road, Shanghai 200444, P. R. China
Yongming Dai: Department of Physics, Shanghai University, 99 ShangDa Road, Shanghai 200444, P. R. China
Yuan Wang: Department of Physics, Shanghai University, 99 ShangDa Road, Shanghai 200444, P. R. China

International Journal of Modern Physics C (IJMPC), 2005, vol. 16, issue 07, 1027-1041

Abstract: 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,$H_0^2\sim 16.28$. By using a proper first-order derivative (FOD) 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 asO (h14); (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 below5×10-15, than the one widely used in the lectures, whose error tolerance is below 10-11.

Keywords: Obrechkoff method; high-order derivative; multistep method; second-order initial value problem with periodic solutions; numerical solution to the Duffing equation (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1142/S0129183105007716

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