2024/09/06 by Alejandro Cabrera, David Martı́n de Diego, Cabrera, Alejandro +3 · 1 citation
Mathematics · #53D05 #65P10 #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2409.04342
openalex publication_date 2024/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
While the construction of symplectic integrators for Hamiltonian dynamics is well understood, an analogous general theory for Poisson integrators is still lacking. The main challenge lies in overcoming the singular and non-linear geometric behavior of Poisson structures, such as the presence of symplectic leaves with varying dimensions. In this paper, we propose a general approach for the construction of geometric integrators on any Poisson manifold based on independent geometric and dynamic sources of approximation. The novel geometric approximation is obtained by adapting structural results about symplectic realizations of general Poisson manifolds. We also provide an error analysis for the resulting methods and illustrative applications.