  We study the convergence in total variation and $V$-norm of discretization
schemes of the underdamped Langevin dynamics. Such algorithms are very popular
and commonly used in molecular dynamics and computational statistics to
approximatively sample from a target distribution of interest. We show first
that, for a very large class of schemes, a minorization condition uniform in
the stepsize holds. This class encompasses popular methods such as the
Euler-Maruyama scheme and the schemes based on splitting strategies. Second, we
provide mild conditions ensuring that the class of schemes that we consider
satisfies a geometric Foster--Lyapunov drift condition, again uniform in the
stepsize. This allows us to derive geometric convergence bounds, with a
convergence rate scaling linearly with the stepsize. This kind of result is of
prime interest to obtain estimates on norms of solutions to Poisson equations
associated with a given numerical method.
