We consider geometric numerical integration algorithms for differential equations evolving on symmetric spaces. The integrators are constructed from canonical operations on the symmetric space, its Lie triple system (LTS), and the exponential from the LTS to the symmetric space. Examples of symmetric spaces are n-spheres and Grassmann manifolds, the space of positive definite symmetric matrices, Lie groups with a symmetric product, and elliptic and hyperbolic spaces with constant sectional curvatures. We illustrate the abstract algorithm with concrete examples. In particular for the n-sphere and the n-dimensional hyperbolic space the resulting algorithms are very simple and cost only O(n) operations per step.

翻译：我们考虑对称空间上演化微分方程的几何数值积分算法。该积分器由对称空间的基本运算、其李三重系统（LTS）以及从LTS到对称空间的指数映射构建而成。对称空间的实例包括n维球面、格拉斯曼流形、正定对称矩阵空间、具有对称乘积的李群以及常截面曲率的椭圆空间与双曲空间。我们通过具体示例阐明该抽象算法。特别地，对于n维球面与n维双曲空间，所得算法极为简洁，每步计算复杂度仅为O(n)次运算。