  Stochastic evolutions of classical field theories have recently become
popular in the study of problems such as determination of the rates of
topological transitions and the statistical mechanics of nonlinear coherent
structures. To obtain high precision results from numerical calculations, a
careful accounting of spacetime discreteness effects is essential, as well as
the development of schemes to systematically improve convergence to the
continuum. With a kink-bearing $\phi^4$ field theory as the application arena,
we present such an analysis for a 1+1-dimensional Langevin system. Analytical
predictions and results from high resolution numerical solutions are found to
be in excellent agreement.
