Butcher series for Hamiltonian Poisson integrators through symplectic groupoids
Adrien Busnot Laurent, Oscar Cosserat
TL;DR
This work develops a cohesive algebraic framework that blends Poisson geometry with numerical analysis by placing Hamiltonian dynamics on Poisson manifolds inside symplectic groupoids. It introduces a pre-Lie algebra structure on jets of Lagrangian bisections and uses Butcher-tree (B-series) methods, together with the Butcher–Connes–Kreimer Hopf algebra, to encode high-order Hamilton-Jacobi flows. The authors then construct Hamiltonian Poisson integrators via two routes: (i) Taylor-series truncations that yield high-order, structure-preserving maps, and (ii) Runge-Kutta–type schemes (RKHP) with explicit order conditions, including degenerate, low-stage variants for certain birealisations. This provides a dual geometric-computational toolkit, linking the groupoid viewpoint of Poisson geometry to practical high-order integrators that stay on symplectic leaves and preserve Casimirs, with potential extensions to stability analysis and backward error studies. The framework thus advances both the theoretical understanding of Lagrangian bisections and the development of robust, high-order numerical methods for Hamiltonian dynamics on Poisson manifolds.
Abstract
We exhibit a new pre-Lie algebra in the framework of symplectic groupoids and, in turn, introduce a pre-Lie formalism of Butcher trees for the approximation of Hamilton-Jacobi solutions on any symplectic groupoid $\mathcal{G} \rightrightarrows M.$ The impact of this new algebraic approach is twofold. On the geometric side, it yields algebraic operations to approximate Lagrangian bisections of $\mathcal{G}$ using the Butcher-Connes-Kreimer Hopf algebra and, in turn, aims at a better understanding of the group of Hamiltonian diffeomorphisms of $M.$ On the computational side, we define a new class of Poisson integrators for Hamiltonian dynamics on Poisson manifolds.