  We derive for the first time in the literature a rate of convergence in the
hydrodynamic limit of the Kawasaki dynamics for a one-dimensional lattice
system. We use an adaptation of the two-scale approach. The main difference to
the original two-scale approach is that the observables on the mesoscopic level
are described by a projection onto splines of second order, and not by a
projection onto piecewise constant functions. This allows us to use a more
natural definition of the mesoscopic dynamics, which yields a better rate of
convergence than the original two-scale approach.
