In order to perform numerical simulations of the Kardar-Parisi-Zhang (KPZ) equation, in any dimensionality, a spatial discretization scheme must be prescribed. The known fact that the KPZ equation can be obtained as a result of a Hopf-Cole transformation applied to a diffusion equation (with multiplicative noise) is shown here to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, the discretization prescriptions for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen. On the other hand, since the discretization is an operation performed on space and the Hopf-Cole transformation is local both in space and time, the former should be the same regardless of the field to which it is applied. It is shown that whereas some discretization schemes pass both consistency tests, known examples in the literature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to be a natural and safe starting point in choosing spatial discretization schemes. We also analyze the relation between real-space and pseudospectral discrete representations. In addition we discuss the relevance of the Galilean-invariance violation in these consistent discretization schemes and the alleged conflict of standard discretization with the fluctuation-dissipation theorem, peculiar of one dimension.

Interfaces of rather different natures—as, e.g., bacterial colony or forest fire boundaries, or semiconductor layers grown by different methods (MBE, sputtering, etc.)—are self-affine fractals, and feature scaling with universal exponents (depending on the substrate’s dimensionality d and global topology, as well as on the driving randomness’ spatial and temporal correlations but not on the underlying mechanisms). Adding lateral growth as an essential (non-equilibrium) ingredient to the known equilibrium ones (randomness and interface relaxation), the Kardar–Parisi–Zhang (KPZ) equation succeeded in finding (via the dynamic renormalization group) the correct exponents for flat $d=1$ substrates and (spatially and temporally) uncorrelated randomness. It is this interplay which gives rise to the unique, non-Gaussian scaling properties characteristic of the specific, universal type of non-equilibrium roughening. Later on, the asymptotic statistics of process $h\left(x\right)$ fluctuations in the scaling regime was also analytically found for $d=1$ substrates. For $d>1$ substrates, however, one has to rely on numerical simulations. Here we review a variational approach that allows for analytical progress regardless of substrate dimensionality. After reviewing our previous numerical results in $d=1$, 2, and 3 on the time evolution of one of the functionals—which we call the non-equilibrium potential (NEP)—as well as its scaling behavior with the nonlinearity parameter $\lambda$, we discuss the stochastic thermodynamics of the roughening process and the memory of process $h\left(x\right)$ in KPZ and in the related Golubović–Bruinsma (GB) model, providing numerical evidence for the significant dependence on initial conditions of the NEP’s asymptotic behavior in both models. Finally, we highlight some open questions.