Magnus And Neumann Expansion Methods For Linear Highly-oscillatory Ordinary Differential Equations

This paper deals with the numerical solutions of highly-oscillatory ordinary differential equations, with a special reference to the linear systems of the form y"+g(t)y=0,where (?) g(t)=+∞.Highly-oscillatory ordinary differential equations are a kind of equations whose solutions are highly-oscillatory. 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, dealling with the linear highly-oscillatory systems y"+g(t)y=0, numerical methods such as classical Runge-Kutta methods,multi-step methods and so on will produce bigger error. Recently, using Magnus expansion Iserles has studied numerical methods for this kind of equations in detail and given good numerical methods.In this paper, we introduce Magnus expansion and Neumann expansion methods systematically. For numerical methods constucted by Neumann expansion, the highly oscillatory integrals are concerned. We compute them with Filon method, Taylor expansion method, and piecewise linear interpolation method, and give different numerical methods. Experimental results show that these methods can give better numerical results.