Construction of an extended invariant for an arbitrary ordinary differential equation with its development in a numerical integration algorithm
Construction of an extended invariant for an arbitrary ordinary differential equation with its development in a numerical integration algorithm
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DOI：
10.1103/physreve.73.026703
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发表时间：
2006-02-01
影响因子：
2.4
通讯作者：
Nakamura, H
中科院分区：
文献类型：
--
作者：
Fukuda, I;Nakamura, H
对于任意的常微分方程，本文提出了一种构造具有时不变函数的扩展常微分方程的方案。这个计划使我们能够检查的精度数值积分的常微分方程，本身可能没有不变量。这些量是通过参考Nose-Hoover分子动力学方程及其相关守恒量构造的。通过将此程序应用于几个分子动力学方程，在每个动力学中单独定义的常规守恒量可以以统一的，广义的方式再现;我们的概念允许这些量和思想的透明前景。发展的技术，对于某一类的常微分方程，我们构建了一个数值积分，不仅是明确的和对称的，但保留了适当定义的扩展常微分方程，这也提供了一个不变的单位雅可比。因此，我们的概念是简单地建立一个无发散的扩展常微分方程，其解决方案只是一个提升的原始常微分方程，并构成一个有效的积分器，保持相空间体积的扩展系统。我们提出精确的讨论积分器的一般数学性质，并提供具体的条件，应纳入实际应用。
For an arbitrary ordinary differential equation (ODE), a scheme for constructing an extended ODE endowed with a time-invariant function is here proposed. This scheme enables us to examine the accuracy of the numerical integration of an ODE that may itself have had no invariant. These quantities are constructed by referring to the Nose-Hoover molecular dynamics equation and its related conserved quantity. By applying this procedure to several molecular dynamics equations, the conventional conserved quantity individually defined in each dynamics can be reproduced in a uniform, generalized way; our concept allows a transparent outlook underlying these quantities and ideas. Developing the technique, for a certain class of ODEs we construct a numerical integrator that is not only explicit and symmetric, but preserves a unit Jacobian for a suitably defined extended ODE, which also provides an invariant. Our concept is thus to simply build a divergence-free extended ODE whose solution is just a lift-up of the original ODE, and to constitute an efficient integrator that preserves the phase-space volume on the extended system. We present precise discussions about the general mathematical properties of the integrator and provide specific conditions that should be incorporated for practical applications.