In this talk, I will give a brief overview of geometric integrators - numerical methods based on the idea of preserving some geometric structures in discretized versions of differential equations. I will focus on some recent results in that context, related to Poisson and Dirac geometry. We will also discuss Dirac structures (not) appearing within the framework of port-Hamiltonian systems and eventually of the systems with constraints. From the mathematical point of view, Dirac structures generalize simultaneously symplectic and Poisson structures. Time permitting, I will also mention an approach to the variational formulation of dynamics on Dirac structures. The talk is based on the following papers:

1. V.Salnikov, A.Hamdouni, From modelling of systems with constraints to generalized geometry and back to numerics, Z Angew Math Mech., Vol. 99, Issue 6, 2019.
2. V.Salnikov, A.Hamdouni, D.Loziienko, Generalized and graded geometry for mechanics: a comprehensive introduction, Mathematics and Mechanics of Complex Systems, Vol. 9, No. 1, 2021.
3. O. Cosserat, C. Laurent-Gengoux, A. Kotov, L. Ryvkin, V. Salnikov, On Dirac structures admitting a variational approach, Mathematics and Mechanics of Complex Systems, 2023.
4. V. Salnikov, Port-Hamiltonian systems: structure recognition and applications, Programming and Computer Software, Volume 50, 2, 2024.
5. V.Salnikov, A.Falaize, D.Loziienko. Learning port-Hamiltonian systems - algorithms, Computational Mathematics and Mathematical Physics, 2023.
6. O.Cosserat, V.Salnikov, C.Laurent-Gengoux, Numerical Methods in Poisson Geometry and their Application to Mechanics, to appear in Mathematics and Mechanics of Solids, 2024.