Abstract

A new numerical method for solving the Cauchy problem for Hamiltonian systems is tested in detail as applied to two benchmark problems: the one-dimensional motion of a point particle in a cubic potential field and the Kepler problem. The global properties of the resulting approximate solutions, such as symplecticity, time reversibility, total energy conservation, and the accuracy of numerical solutions to the Kepler problem are investigated. The proposed numerical method is compared with three-stage symmetric symplectic Runge–Kutta methods, the discrete gradient method, and nested implicit Runge–Kutta methods.

中文翻译：

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测试一种新的保守方法来解决测试问题上的哈密顿系统的柯西问题

摘要

详细测试了一种新的求解哈密顿系统柯西问题的数值方法，并将其应用于两个基准问题：立方势场中点粒子的一维运动和开普勒问题。研究了所得近似解的全局性质，例如辛性，时间可逆性，总能量守恒以及开普勒问题数值解的准确性。将所提出的数值方法与三阶段对称辛Runge–Kutta方法，离散梯度方法和嵌套隐式Runge–Kutta方法进行了比较。