  We develop renormalization group methods for solving partial and stochastic
differential equations on coarse meshes. Renormalization group transformations
are used to calculate the precise effect of small scale dynamics on the
dynamics at the mesh size. The fixed point of these transformations yields a
perfect operator: an exact representation of physical observables on the mesh
scale with minimal lattice artifacts. We apply the formalism to simple
nonlinear models of critical dynamics, and show how the method leads to an
improvement in the computational performance of Monte Carlo methods.
