  In this paper, we analyze the preservation of asymptotic properties of
partially dissipative hyperbolic systems when switching to a discrete setting.
We prove that one of the simplest consistent and unconditionally stable
numerical methods - the central finite difference scheme - preserves both the
asymptotic behaviour and the parabolic relaxation limit of one-dimensional
partially dissipative hyperbolic systems which satisfy the Kalman rank
condition.
  The large time asymptotic-preserving property is achieved by conceiving
time-weighted perturbed energy functionals in the spirit of the hypocoercivity
theory. For the relaxation-preserving property, drawing inspiration from the
observation that solutions in the continuous case exhibit distinct behaviours
in low and high frequencies, we introduce a novel discrete Littlewood-Paley
theory tailored to the central finite difference scheme. This allows us to
prove Bernstein-type estimates for discrete differential operators and leads to
a new relaxation result: the strong convergence of the discrete linearized
compressible Euler system with damping towards the discrete heat equation,
uniformly with respect to the mesh parameter.
