  In a recent paper by Grunewald et.al., a new method to study hydrodynamic
limits was developed for reversible dynamics. In this work, we generalize this
method to a family of non-reversible dynamics. As an application, we obtain
quantitative rates of convergence to the hydrodynamic limit for a weakly
asymmetric version of the Ginzburg-Landau model endowed with Kawasaki dynamics.
These results also imply local Gibbs behavior, following a method introduced in
a recent paper by the second author.
