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 quartic scalar field theory as the application arena, we present such an analysis for a 1+1-dimensional Langevin system. High resolution numerical results provide excellent supporting evidence for our analytical predictions.