In this paper we propose a geometric integrator to numerically approximate the flow of Lie systems. The highlight of this paper is to present a novel procedure that integrates the system on a Lie group intrinsically associated to the Lie system, and then generating the discrete solution of this Lie system through a given action of the Lie group on the manifold where the system evolves. One major result from the integration on the Lie group is that one is able to solve all automorphic Lie systems at the same time, and that they can be written as first-order systems of linear homogeneous ODEs in normal form. This brings a lot of advantages, since solving a linear ODE involves less numerical cost. Specifically, we use two families of numerical schemes on the Lie group, which are designed to preserve its geometrical structure: the first one based on the Magnus expansion, whereas the second is based on RKMK methods. Moreover, since the aforementioned action relates the Lie group and the manifold where the Lie system evolves, the resulting integrator preserves any geometric structure of the latter. We compare both methods for Lie systems with geometric invariants, particularly a class on Lie systems on curved spaces. As already mentioned, the milestone of this paper is to show that the method we propose preserves all the geometric invariants very faithfully, in comparison with nongeometric numerical methods.

翻译：本文提出一种几何积分器，用于数值逼近李系统的流。本文亮点在于提出一种新流程：首先在与李系统本质相关的李群上积分该系统，然后通过李群在系统演化流形上的给定作用生成该李系统的离散解。李群积分的一个重要成果是能够同时求解所有自守李系统，且这些系统可写成标准形式的一阶线性齐次常微分方程组。这带来诸多优势，因为求解线性常微分方程的计算代价较低。具体而言，我们采用两类保持李群几何结构的数值格式：第一类基于马格努斯展开，第二类基于RKMK方法。此外，由于前述作用建立了李群与李系统演化流形间的关联，所得积分器能保持后者的任意几何结构。我们比较了两种方法在具有几何不变量的李系统（特别是曲率空间上的李系统）中的表现。如前所述，本文的里程碑式成果在于证明：与非几何数值方法相比，我们所提出的方法能极精确地保持所有几何不变量。