Numerical integration of ordinary differential equations on manifolds
Numerical integration of ordinary differential equations on manifolds
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DOI：
10.1007/bf02429858
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发表时间：
1993-12
影响因子：
3
通讯作者：
P. Crouch;R. Grossman
中科院分区：
文献类型：
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作者：
P. Crouch;R. Grossman
本文研究微分方程数值积分算法的发展问题，当微分方程被看作是在某些欧氏空间中的方程时，自然地在某些嵌入的子流形上演化。希望构造其迭代也在同一流形上演化的算法。因此，这些算法可以被视为集成常微分方程流形上。基本方法将子流形上流动的计算从数值积分过程中“简化”出来。利用“冻结”由定义向量场得到的微分算子系数的概念，可以提出并从理论上分析两类单步和多步算法。显式三阶算法推导出，与额外的方程增加了他们的经典同行，从“障碍物”定义的非零李括号。
This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.