We propose a geometric integrator to numerically approximate the flow of Lie systems. The key is a novel procedure that integrates the Lie system on a Lie group intrinsically associated with a Lie system on a general manifold via a Lie group action, and then generates the discrete solution of the Lie system on the manifold via a solution of the Lie system on the Lie group. One major result from the integration of a Lie system on a Lie group is that one is able to solve all associated Lie systems on manifolds at the same time, and that Lie systems on Lie groups can be described through first-order systems of linear homogeneous ordinary differential equations (ODEs) in normal form. This brings a lot of advantages, since solving a linear system of ODEs 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 Runge-Kutta-Munthe-Kaas (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. We also illustrate the superiority of our method for describing long-term behavior and for differential equations admitting solutions whose geometric features depends heavily on initial conditions. As already mentioned, our milestone 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。其核心是一个新方法：通过李群作用，在与一般流形上的李系统内在关联的李群上积分该李系统，进而通过李群上的李系统解生成流形上李系统的离散解。李群上李系统积分的重要成果在于：可同时求解流形上所有关联李系统，且李群上的李系统可通过规范形式的一阶线性齐次常微分方程组描述。这带来诸多优势——线性常微分方程组的求解计算成本更低。具体而言，我们在李群上采用两类保持几何结构的数值格式：一类基于Magnus展开，另一类基于Runge-Kutta-Munthe-Kaas（RKMK）方法。由于前述作用关联了李系统演化的李群与流形，所构建的积分器能够保持流形的任意几何结构。我们通过几何不变量（尤其是弯曲空间上的李系统类）对两类方法进行比较，并展示了本方法在描述长期行为及处理解几何特征高度依赖初始条件的微分方程方面的优越性。正如前文所述，关键突破在于：相较于非几何数值方法，本方法能够极其忠实地保持所有几何不变量。