An improved Störmer-Verlet method based on exact discretization for nonlinear oscillators

Abstract

• A modified Störmer-Verlet method with a parameter ω is proposed. • The modified method is symplectic, symmetric, convergent with order two and explicit for separable Hamiltonian system. • The modified method solves exactly some linear equation associated to a nonlinear oscillator equation. • Choosing ω in modified method by approximate frequency of nonlinear oscillator is much more accurate than the classical Störmer-Verlet method. • Results are verified by numerical experiments on the cubic Duffing and simple pendulum equation. Motivated by the advantage of exact discretization of a linear differential equation and the importance of symplectic numerical methods for conservative nonlinear oscillators, a modified Störmer-Verlet method relying on a parameter ω is proposed. The main idea is: firstly, based on some analytical approximation strategies, relating a linear equation with the corresponding nonlinear equation such that linear equation’s frequency approximates the exact frequency of the nonlinear equation; secondly, forcing the modified Störmer-Verlet method to solve the related linear equation exactly. The convergence, symplectic and symmetric properties of the new method are analyzed. For numerical implementation, the cubic Duffing equation and the simple pendulum are solved by the new method with some approximate frequencies as the parameter ω , respectively. Numerical results show that the new method is much more accurate than its classical partner.