In my talk, I shall explore a possible approach to
constructing a geometric integrator in the context of contact geometry.
Generally speaking, geometric integrators are numerical algorithms that
preserve the geometric structure of the underlying differential
equations, making them particularly useful for long-term numerical
integration. The first part of the talk will be devoted to introducing
these integrators. Subsequently, I shall review the necessary concepts
of contact geometry, with a particular focus on the novel approach using
symplectic homogeneous manifolds. The symplectization of a contact
manifold is especially useful in numerical contexts, as it enables the
application of well-established symplectic integrators. I will present
this construction in detail and discuss potential challenges. Lastly, I
shall thoroughly illustrate this approach to contact integrators through
two examples: the damped harmonic oscillator and the dynamics on a
Moebius strip. This talk is based on a project I conducted under the
supervision of  Katarzyna Grabowska.