  We demonstrate and explain that conventional finite difference schemes for
direct numerical integration do not approximate the continuum
Kardar-Parisi-Zhang (KPZ) equation due to microscopic roughness. The effective
diffusion coefficient is found to be inconsistent with the nominal one. We
propose a novel discretization in 1+1 dimensions which does not suffer from
this deficiency and elucidates the reliability and limitations of direct
integration approaches.
