  We consider time discretizations of the Vlasov-HMF (Hamiltonian Mean-Field)
equation based on splitting methods between the linear and non-linear parts. We
consider solutions starting in a small Sobolev neighborhood of a spatially
homogeneous state satisfying a linearized stability criterion (Penrose
criterion). We prove that the numerical solutions exhibit a scattering behavior
to a modified state, which implies a nonlinear Landau damping effect with
polynomial rate of damping. Moreover, we prove that the modified state is close
to the continuous one and provide error estimates with respect to the time
stepsize.
