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Fecha
2023-12-23
Derechos de acceso
info:eu-repo/semantics/openAccess
Título de la revista
ISSN de la revista
Título del volumen
Editorial
Springer
Citas
Resumen
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.
Descripción
The registered version of this article, first published in Journal of Nonlinear Science, is available online at the publisher's website: Springer, https://doi.org/10.1007/s00332-023-10000-8
La versión registrada de este artículo, publicado por primera vez en Journal of Nonlinear Science, está disponible en línea en el sitio web del editor: Springer, https://doi.org/10.1007/s00332-023-10000-8
La versión registrada de este artículo, publicado por primera vez en Journal of Nonlinear Science, está disponible en línea en el sitio web del editor: Springer, https://doi.org/10.1007/s00332-023-10000-8
Categorías UNESCO
Palabras clave
Geometric methods in ODEs, Discrete differential geometry, Discretization methods and integrators
Citación
Blanco, Jiménez, de Lucas, & Sardón. (2024). Geometry-Preserving Numerical Methods for Physical Systems with Finite-Dimensional Lie Algebras. Journal of Nonlinear Science, 34(1). https://doi.org/10.1007/S00332-023-10000-8
Centro
E.T.S. de Ingenieros Industriales
Departamento
Matemática Aplicada I