A fixed time-step variational integrator cannot preserve momentum, energy, and symplectic form simultaneously for nonintegrable systems. This barrier can be overcome by treating time as a discrete dynamic variable and deriving adaptive time-step variational integrators that conserve the energy in addition to being symplectic and momentum-preserving. Their utility, however, is still an open question due to the numerical difficulties associated with solving the governing discrete equations. In this work, we investigate the numerical performance of energy-preserving, adaptive time-step variational integrators. First, we compare the time adaptation and energy performance of the energy-preserving adaptive algorithm with the adaptive variational integrator for Kepler's problem. We also study the effect of variable precision arithmetic on the energy conservation properties. Second, we apply tools from Lagrangian backward error analysis to investigate numerical stability of the energy-preserving adaptive algorithm. Finally, we consider a simple mechanical system example to illustrate our backward stability approach by constructing a modified Lagrangian for the modified equation of an energy-preserving, adaptive time-step variational integrator.
翻译：固定的时步变异整合器无法同时为非可分解系统保留动力、 能量和间位变异形式。 这个屏障可以通过将时间作为离散的动态变量和产生适应性的时间步变异整合器来克服,这些变异整合器不仅能够保护能源,而且还能保护能源。 但是,由于解决管理离散方程式的数值困难,它们的效用仍然是一个未决问题。 在这项工作中,我们调查了节能和适应性时间步变异整合器的数值性能。 首先,我们将节能适应性算法的时间适应和能量性能与开普勒问题的适应性变异化集成器进行了比较。 我们还研究了可变精确算法对节能特性的影响。 其次,我们运用了拉格朗日落后错误分析工具来调查节能适应性算法的数字稳定性。 最后,我们考虑一个简单的机械系统示例,通过为节能、 适应性时序变形变形的化方程式建造一个修改的拉格朗格来说明我们的后向稳定性方法。
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