Magnus Expansion Methods For Highly-oscillatory Differential Equations

Highly-oscillatory differential equations are a kind of equations whose solutions are highly-oscillatory. It is extensively applied in molecular dynamics, celestial mechanics, quantum chemistry, atomic physics and so on. Therefore, it is significant to study its numerical methods.It is very difficult to give a good numerical method for highly-oscillatory ordinary differential equations. Recently, using Magnus expansion Iserles has studied numerical methods which can deal with the linear highly-oscillatory systems y"+g(t)y=0 in detail and given good numerical methods.In this paper, we introduce the properties of Hamiltonian equations, symplectic geometric algorithms, Magnus expansion, modified Magnus expansion and Neumann expansion methods. We mainly discuss a kind of highly-oscillatory differential equations which take the form Y'+AY=B(t,Y)Y. First, using Picard iteration method we can get modified Neumann expansion form of linear highly-oscillatory differential equations. Then, we give a numerical method to deal with these equations by using modified Magnus expansion method. For the numerical methods which we construct concerns the highly oscillatory integrals, we compute them with Filon method, and piecewise linear interpolation method. And then we give different numerical methods. Experimental results show that these methods can give better numerical results. Finally, we promote this method to deal with nonlinear problems. For example, the FPU problems can be writen as this kind of equations. We consider modified Magnus expansion methods for this problem.