The Study On Continuous-stage Methods For Second-order Oscillatory Differential Equations

Highly oscillatory differential equations often arise in different fields such as astronomical mechanics,theoretical physics,chemistry and molecular biology,etc,and there is a growing concern about numerial solution in recent years.On the other hand,in some cases the differential equation is often equivalent to a Hamilton system with energy conservation and symplectic geometry structure.One of the important ideas of modern computing methods is to maintain the essential characteristics of the problem being studied.Therefore,it is of great significance for the numerical methods of oscillatory differential equations to study oscillatory characteristics,energy preservation and symplectic geometry structure.Recently,continuous-stage methods for preserving energy have been developed.In this paper,based on the work of predecessors,a continuous-stage Runge-Kutta-Nystr(?)m method of trigonometric fitting for the second order oscillation differential equation is proposed and its energy-preserving properties are studied;A specific symmetric and energy-preserving trigonometrically fitted continuous-stage RungeKutta-Nystr(?)m methods are constructed.Numerical experiments show that the method described in this paper can well describe the oscillating structure of the problem and accurately maintain the energy of the system.