Numerical integration of KPZ equation with restrictions

Abstract

In this paper, we introduce a novel integration method of Kardar-Parisi-Zhang (KPZ) equation. It has always been known that if during the discrete integration of the KPZ equation the nearest-neighbor height-difference exceeds a critical value, an instability appears and the integration diverges. One way to avoid these instabilities is to replace the KPZ nonlinear-term by a function of the same term that depends on a single adjustable parameter which is able to control pillars or grooves growing on the interface. Here, we propose a different integration method which consists of directly limiting the value taken by the KPZ nonlinearity, thereby imposing a restriction rule that is applied in each integration time-step, as if it were the growth rule of a restricted discrete model, e.g. restricted-solid-on-solid (RSOS). Taking the discrete KPZ equation with restrictions to its dimensionless version, the integration depends on three parameters: the coupling constant $g$, the inverse of the time-step $k$, and the restriction constant $\varepsilon$ which is chosen to eliminate divergences while keeping all the properties of the continuous KPZ equation. We study in detail the conditions in the parameters' space that avoids divergences in the 1-dimensional integration and reproduce the scaling properties of the continuous KPZ with a particular parameter set. We apply the tested methodology to the $d$-dimensional case ($d = 3,4$) with the purpose of obtaining the growth exponent $\beta$, by establishing the conditions of the coupling constant $g$ under which we recover known values reached by other authors, in particular for the RSOS model. This method allows us to infer that $d = 4$ is not the critical dimension of the KPZ universality class, where the strong-coupling phase dissapears.