The Predictor-Corrector of Exponential Time Differencing forStrongly Nonlinear Autonomous Oscillators with Many DOFs

摘要

The predictor-corrector methods of exponential time differencing are extended to determine thebifurcations and limit cycles of strongly nonlinear autonomous oscillators with many degrees offreedom.The numerical method treated in this paper is the so-called "exponential time differencing" (ETD)scheme [1, 2, 3], or called "exponential fitting"(EF) [4, 5]. In [6] the various order explicit multistepexponential time differencing was developed which works extremely well for nonlinear systems. Inthis paper, we use explicit multistep exponential time differencing to obtain the predicted solutionvalue, then use these predicted values in the corresponding implicit method to obtain thecorrected solution value. This combination of an explicit ETD and implicit ETD scheme iscalled predictor-corrector ETD method. We test the predictor-corrector ETD method on coupledvan der Pol oscillators and coupled Rayleigh oscillators, and compare with the well studied andwidely used four-order Runge-Kutta method. Limit cycles of the oscillators can be calculated toand desired degree of accuracy. The numerical results show that the schemes are highly accurateand computationally efficient.