Symmetric Numerical Methods For Highly-oscillatory Differential Equations

Highly-oscillatory differential equations are a kind of equations whose solutions are highly-oscillatory, which are extensively applied in molecular dynamics, celestial mechanics, quantum chemistry, atomic physics and so on. Therefore, it is signi--ficant to study its numerical methods .For highly-oscillatory differential equations, however, it is hard to give good computational results with general numerical methods.A basic idea behind the design of the numerical schemes is that they can preserve the properties of the original problems as much as possible. Hamiltonian functions (which usually mean energy) are conservative quantities of Hamiltonian equations. According to the above principle, we also expect that numerical methods could keep this property better.In this paper, we systematically introduce the properties of Hamiltonian equations, symplectic geometric algorithms and symmetric, composition and splitting numerical methods. We mainly discuss a kind of highly-oscillatory differential equations whichtake the form (?).This kind of equations can be written asHamiltonian equations, and the corresponding Hamiltonian functions are conservative quantities. The FPU problems can be expressed as the form of this kind of equations. We give two new symmetric numerical schemes for the equations. The numerical experiment results for FPU problems show that two symmetric numerical methods have better behavior of energy conservation.