Study For Numerical Methods Of Highly-oscillatory Differential Equations

Highly-oscillatory differential equations are a kind of equations whose solutions are highly-oscillatory,it is extensively applied in aspects such as molecular dynamics,celestial mechanics,quantum chemistry,atomic physics and so on.Therefore, it is significant to study its numerical methods.For highly-oscillatory differential equations,it is hard to give good computational results with general numerical methods.For example,dealling with the linear highly-oscillatory systems y"+g(t)y=0,classical numerical methods 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.E.Hairer et al have studied symmetric numerical methods for highly-oscillatory differential equations.In this paper,we introduce the properties of Hamiltonian equations,symplectic geometric algorithms and symmetric,Magnus expansion and Neumann expansion methods.Iteration plays an important role in Magnus expansion and Neumann expansion methods,using iterative method we consinder numerical methods for highly-oscillatory differential equations.The numerical experiment results for FPU problems show that the method which based on iteration can give better numerical results.