A discrete-cell formulation of hydrodynamics was recently introduced, which is exactly renormalizable in a certain sense: if one knows the discrete equations of motion for a certain cell size W and discrete time interval τ, one can accurately numerically calculate the equations of motion on the coarser scales 2W or 2τ. These coarsening transformations have previously been investigated for the one-dimensional diffusive system. A line of fixed points was found, parameterized by the (positive) diffusivity D'. In this paper we examine the behavior of the coarsening transformation on the D' = 0 manifold in the space of equations of motion for one-dimensional systems. We find another line of fixed points, this one parameterized by the super-Burnett coefficient D'3. This corresponds to a Gaussian critical point. The possibility of generalizing this to non-Gaussian (Ising-like) critical points is discussed.
