Collocation and Relaxed Collocation for the Fer and the Magnus Expansions
Collocation and Relaxed Collocation for the Fer and the Magnus Expansions
复制标题
DOI：
10.1137/s0036142997326616
复制
发表时间：
1999-04
影响因子：
2.9
通讯作者：
A. Zanna
中科院分区：
文献类型：
--
作者：
A. Zanna
本文考虑非线性矩阵李群ODE y' = \gamma（t，y）y$，$y（0）= y_0，$数值解的Fer展开和Magnus展开，其中y在矩阵李群G中演化，$\gamma（t，y）$在李代数g中. $\ghe$.从区分群中执行的运算和切空间中执行的运算的几何方法出发，我们构造了基于搭配的李群不变量方法。我们证明了，只要两个展开式被正确截断，配置节点c_1，c_2，\ldots，c_{\nu}$产生的数值方法的阶数与经典情形相同.我们也放松配置条件，从而设计显式方法的顺序3。最后，我们讨论了所提出的方法在数值实验中产生的哈密顿力学，投影技术的结果进行比较。
We consider the Fer and the Magnus expansions for the numerical solution of the nonlinear matrix Lie-group ODE $y' = \gamma(t,y) y$, $y(0)= y_0,$ where y evolves in a matrix Lie group G and $\gamma(t,y)$ is in the Lie algebra %$\hbox{\Fr{g}}$. $\ghe$. Departing from a geometrical approach that distinguishes between those operations performed in the group and those performed in the tangent space, we construct Lie-group invariant methods based on collocation. We prove that, as long as the two expansions are correctly truncated, the collocation nodes $c_1, c_2, \ldots, c_{\nu}$ yield numerical methods whose order is the same as in the classical setting. We also relax the collocation conditions, thereby devising explicit methods of order 3. To conclude, we discuss the proposed methods in a numerical experiment that arises in Hamiltonian mechanics, comparing the results with projection techniques.