Geometry-Preserving Numerical Methods for Physical Systems with Finite-Dimensional Lie Algebras

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2023-12-23 DOI:10.1007/s00332-023-10000-8

L. Blanco, F. Jiménez, J. de Lucas, C. Sardón

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Abstract

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 is 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 non-geometric numerical methods.

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有限维李代数物理系统的几何保全数值方法

摘要我们提出了一种几何积分器，用于对列系的流进行数值逼近。其关键在于一个新颖的程序，它通过一个李群作用将一个李群上的李系与一个一般流形上的李系内在地联系在一起，然后通过一个李群上的李系的解生成流形上的李系的离散解。将一个 Lie 系统集成到一个 Lie 群上的一个主要结果是，人们能够同时求解流形上所有相关的 Lie 系统，并且 Lie 群上的 Lie 系统可以通过正则形式的一阶线性均相常微分方程（ODE）系统来描述。这带来了很多好处，因为求解线性 ODE 系统的数值代价较低。具体地说，我们使用了两组旨在保留李群几何结构的数值方案：第一组基于马格努斯展开，而第二组则基于 Runge-Kutta-Munthe-Kaas (RKMK) 方法。此外，由于上述作用关系到李系演化的李群和流形，由此产生的积分器保留了后者的任何几何结构。我们比较了这两种方法对具有几何不变式的李系的影响，特别是对曲线空间上的一类李系的影响。我们还说明了我们的方法在描述长期行为和微分方程解（其几何特征在很大程度上取决于初始条件）方面的优越性。如前所述，我们的目标是证明，与非几何数值方法相比，我们提出的方法能非常忠实地保留所有几何不变式。

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