Numerical Integration of Lie-Poisson Systems While Preserving Coadjoint Orbits and Energy
Numerical Integration of Lie-Poisson Systems While Preserving Coadjoint Orbits and Energy
复制标题
保持共交轨道和能量的李泊松系统的数值积分
DOI：
--
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发表时间：
2001
影响因子：
2.9
通讯作者：
S. Faltinsen
中科院分区：
文献类型：
--
作者：
Kenth Engø;S. Faltinsen
本文将RKMK型几何积分器应用于Lie-Poisson方程组的数值积分问题。利用李群G在对偶李代数{mbox{
ormalsize$mathfrak{g}$}}^*$为了推进数值流，我们设计了自动停留在余伴随轨道上的任意阶的方法。被称为Casimirs的第一积分通过数值算法保留到机器精度。在所提出的类的方法，我们发现积分器，也保存的能量。这些格式是隐式的二阶格式。讨论了李代数中的非线性迭代和全局误差的线性增长。刚体和有限维截断的二维（2D）不可压缩流体的欧拉方程的数值实验被用来说明该算法的性能。
In this paper we apply geometric integrators of the RKMK type to the problem of integrating Lie--Poisson systems numerically. By using the coadjoint action of the Lie group $G$ on the dual Lie algebra ${mbox{
ormalsize$mathfrak{g}$}}^*$ to advance the numerical flow, we devise methods of arbitrary order that automatically stay on the coadjoint orbits. First integrals known as Casimirs are retained to machine accuracy by the numerical algorithm. Within the proposed class of methods we find integrators that also conserve the energy. These schemes are implicit and of second order. Nonlinear iteration in the Lie algebra and linear error growth of the global error are discussed. Numerical experiments with the rigid body and a finite-dimensional truncation of the Euler equations for a two-dimensional (2D) incompressible fluid are used to illustrate the properties of the algorithm.