VII. SOME NUMERICAL RESULTS We present here some results obtained by numerically integrating the KPZ equation in 1D. Our aim is to compare the standard discretization scheme [Eq. (47)] with the consistent ones presented in Eqs. (10) and (53). To solve Eq. (1) we discretize h ( x, t ) along the substrate direction x with lattice spacing a = 1. We employ a second-order Runge–Kutta algorithm (see e.g. [34]) with periodic boundary conditions. Then the equation of motion is integrated according to the recursive relation with where u$_{j}$ is a Gaussian random variable. Without loss of generality, the interface dynamics can be described in terms of the di- mensionless parameter ˜ λ = (2 ε/ν $^{3}$)1 / $^{2}$λ . In practice, we set ν = ε = 1 and allow λ to vary. The numerical results show that the interface scaling does not depend on the discretization scheme. As shown in Fig. 1, the dynamics fit into the KPZ universality class for all the discretization schemes. The global width, that concerns the fluctuations of the growth height around its mean value, scales according to the Family–Vicsek Ansatz [3] as where the scaling function f ( u ) is defined as On the other hand, correlations can be analyzed in the reciprocal space by means of the structure factor 17 S$_{k}$ ( t ) = 〈 ̂ h$_{k}$ ( t ) ̂ h − $_{k}$( t ) 〉 , (60) f ( u ) ∼     const if u ≪ 1 , u − β otherwise . (59) W ( L, t ) = t $^{β}$f ( t/t$_{x}$ ) = t $^{β}$f ( t/L $^{z}$) , (58) g$_{1}$ = F ( h$_{j}$ ) g$_{2}$ = F ( h$_{j}$ + Δ tg$_{1}$ + (Δ t ) 1 / 2 + u$_{j}$ ) , h$_{j}$ ( t + Δ t ) = h$_{j}$ + Δ t 2 ( g$_{1}$ + g$_{2}$ ) + (Δ t ) 1 / $^{2}$u$_{j}$ , (57) ˙ h$_{j}$ = ν L ( h$_{j}$ ) + λ 2 Q ( h$_{j}$ ) + ε ξ$_{j}$ = F ( h$_{j}$ ) + ε ξ$_{j}$ (56)