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Geometry preserving numerical methods for physical systems with finite-dimensional Lie algebras

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

TL;DR

The paper tackles the challenge of numerically integrating Lie systems while preserving their geometric structure on manifolds. It introduces a geometry-preserving integrator built from a Lie-group reduction: solve a related automorphic Lie system on a Lie group and transfer the solution to the original manifold via a Lie group action, using Magnus expansion or Runge-Kutta-Munthe-Kaas schemes. The key contribution is a concrete 7-step procedure that ensures long-term fidelity and invariants, demonstrated on curved CK spaces where invariants are preserved and qualitative behavior with respect to initial conditions is maintained. This approach offers a principled alternative to nongeometric integrators, with potential impact on simulations where geometric invariants and stratifications are crucial for accurate modeling.

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 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.

Geometry preserving numerical methods for physical systems with finite-dimensional Lie algebras

TL;DR

The paper tackles the challenge of numerically integrating Lie systems while preserving their geometric structure on manifolds. It introduces a geometry-preserving integrator built from a Lie-group reduction: solve a related automorphic Lie system on a Lie group and transfer the solution to the original manifold via a Lie group action, using Magnus expansion or Runge-Kutta-Munthe-Kaas schemes. The key contribution is a concrete 7-step procedure that ensures long-term fidelity and invariants, demonstrated on curved CK spaces where invariants are preserved and qualitative behavior with respect to initial conditions is maintained. This approach offers a principled alternative to nongeometric integrators, with potential impact on simulations where geometric invariants and stratifications are crucial for accurate modeling.

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 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.
Paper Structure (12 sections, 3 theorems, 122 equations, 1 figure, 1 algorithm)

This paper contains 12 sections, 3 theorems, 122 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Geometric fundamentals
  3. Lie groups and matrix Lie groups
  4. Generalities on Lie systems
  5. Automorphic Lie systems
  6. Numerical methods on matrix Lie groups
  7. Numerical methods and automorphic Lie systems
  8. Numerical integration on curved spaces
  9. A class of Lie systems on curved spaces
  10. Numerical integration of a particular example
  11. Initial conditions, long term behavior, and numerical methods
  12. Conclusions and outlook

Key Result

Proposition 2.1

Given a Lie group $G$ and a particular solution $g(t)$ of the Lie system defined on $G$, as where $b_1(t),\ldots,b_r(t)$ are arbitrary $t$-dependent functions and $X_1^\mathrm{R},\ldots,X_r^\mathrm{R}$ are right-invariant vector fields, we have that $g(t) h$ is also a solution of LiesysLiegroup for each $h \in G$.

Theorems & Definitions (6)

  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.2: The 7 step method: Reduction procedure to automorphic Lie systems
  • Theorem 3.1: Magnus, 1954
  • Definition 4.1