2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/151909We 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.14 pages, 3 figures, RevTexStatistical Mechanicstext