天体轨道长期数值积分的误差估计方法

The Long-term Error Estimation Method for the Numerical Integrations of Celestial Orbits

数值积分方法是进行天体力学研究的重要工具,尤其对于行星历表的研究工作而言.由于在使用数值方法计算天体轨道时,最终误差通常是难以预知的,所以在面对精度要求较高或者积分时间较长的工作时具体积分方案的设计---尤其是当使用定步长方法时的步长选择---需要十分谨慎,因为这将意味着是否能在时间成本可以被接受的范围内使解的精度达到要求.因此,在使用数值方法解决实际问题时如何快速寻找效率与精度之间的最佳平衡点是每一个数值积分方法的设计者与使用者都会面临的难题.为解决这一问题,在定步长条件下对数值积分方法的舍入误差概率分布函数以及截断误差积累量对步长的依赖关系和随时间的增长关系进行了深入研究.基于所得结论,提出了一种仅需较少的数值实验资料即可对选择任意时间步长积分至任意积分时刻时的舍入误差概率分布函数与截断误差积累量进行准确估计的方法,并使用Adams-Cowell方法对该误差估计方法在圆周期轨道条件下进行了验证.该误差估计方法在未来有望用于不同数值算法的性能对比研究,同时也可以对数值积分方法求解实际轨道问题时的决策工作带来重要帮助.

Numerical methods have become a very important type of tool for celestial mechanics,especially in the study of planetary ephemerides.The errors generated during the computation are hard to know beforehand when applying a certain numerical integrator to solve a certain orbit.In that case,it is not easy to design a certain integrator for a certain celestial case when the requirement of accuracy were extremely high or the time-span of the integration were extremely large.Especially when a fixed-step method is applied,the caution and effort it takes would always be tremendous in finding a suitable time-step,because it is about whether the accuracy and time-cost of the final result are acceptable.Thus,finding the best balance between efficiency and accuracy with the least time cost appeared to be a major obstruction in the face of both numerical integrator designers and their users.To solve this problem,we investigate the variation pattern of truncation errors and the pattern of rounding error distributions with time-step and time-span of the integration.According to those patterns,we promote an error estimation method that could predict the distribution of rounding errors and the total truncation errors with any time-step at any time-spot with little experimental cost,and test it with the Adams-Cowell method in the calculation of circular periodic orbits.This error estimation method is expected to be applied to the comparison of the performance of different numerical integrators,and also it can be of great help for finding the best solution to certain cases of complex celestial orbits calculations.