Improved discretization of the Kardar-Parisi-Zhang equation

Abstract

We propose a spatial discretization of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions. The exact steady state probability distribution of the resulting discrete surfaces is explained. The effective diffusion coefficient, nonlinearity, and noise strength can be extracted from three correlators, and are shown to agree exactly with the nominal values used in the discrete equations. Implications on the conventional method for direct numerical integration of the KPZ equation are discussed.