Butcher series for Hamiltonian Poisson integrators through symplectic groupoids

We exhibit a new pre-Lie algebra in the framework of symplectic groupoids and, in turn, use Butcher trees to compute the Taylor series of a solution of a Hamilton-Jacobi equation on any symplectic groupoid $\mathcal{G} \rightrightarrows M.$ The impact of this new algebraic approach is twofold. On th