流形上微分方程的算法

Algorithms of Differential Equations on Manifolds

流形上微分方程的数值方法,是近十多年发展起来的、当前热门的数值方法．特别是所谓的的李群方法是在动力学问题的需求下诞生的,它能在弯曲的空间中进行离散化,根本不会出现‘违约问题’．人们将它应用到动力学方程模型等,取得了不少成果,应用前景看好．它的出现可以说是20世纪数值数学领域的新成就．本文主要介绍这一新的理论,并提出了许多急待解决的新的课题．

Algorithms of differential equations on manifolds, are developing hot topics in recent ten years more. Especially, occurrence of Lie group methods associated with wants of the dynamics, it can be used to discretion of the equations in spaces of curvature, and it cann＇t have the drift off the manifold of solution. When it is used to dynamic equations, a lot of well results are obtained, and it is promising methods. Its birthday is a new achievement of the numerical mathematics in the Twenty Century. In this paper, we discuss the new theory, and propose some problems to be solved.