Abstract 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 Stormer-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 Stormer-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.

中文翻译：

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一种改进的基于非线性振荡器精确离散化的 Störmer-Verlet 方法

摘要 基于线性微分方程精确离散化的优点和辛数值方法对保守非线性振子的重要性，提出了一种基于参数ω的改进Stormer-Verlet方法。主要思想是：首先，基于一些解析逼近策略，将线性方程与相应的非线性方程相关联，使得线性方程的频率接近非线性方程的准确频率；其次，强制修正的Stormer-Verlet方法精确求解相关线性方程组。分析了新方法的收敛性、辛性和对称性。对于数值实现，三次Duffing方程和单摆采用新方法求解，以一些近似频率为参数ω，分别。数值结果表明，新方法比其经典方法准确得多。