| This paper deals with the numerical solutions of highly-oscillatory ordinary differential equations,with a special reference to the linear oscillatory" + g(t)y = 0,where lim g(t) = +∞.Highly-oscillatory ordinary differential equations are a kind of equations whose solutions include highly-oscillating functions. They are extensively applied in molecular dynamics, celestial mechanics, quantum chemistry, atomic physics and so on. It is very difficult to give a good numerical method for highly-oscillatory ordinary differential equations. For example, when handling with the linear oscillator y" + g(t)y = 0, with classical methods Runge-Kutta, multi-step methods, these methods may generate bigger error. Recently,Iserles has studied specifically numerical solution question of this kinds of equations using Magnus expansion and given good results.In this paper, we first introduce some basic concepts and knowledge preparing for the following content,and then give a numerical solution for the linear oscillator y" + g(t)y = 0 with trapezoidal method. Both theoretical analysis and numerical experiment indicate that this method generates bigger error. We modify the trapezoidal scheme in several ways. Both error analysis and numerical results indicate that the numerical solutions of modified trapezoidal schemes are better than the trapezoidal scheme.Also, we introduce systematically Magnus methods and modified Magnus methods. Beginning with modified Magnus methods,We consider numerical method of the linear highly-oscillatory ordinary differential based on the Cayley map.The method refers to the highly-oscillating integral.We adopt Filon method ,numerical result indicates that the method has good ability of long time intervals tracking computation.In addition,we compare the numerical results of highly-oscillating integral with Filon and Gauss method separately by numerical experiment.The experiment indicates that Filon method is better than Gauss method.
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