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In order to perform numerical simulations of the KPZ equation, in anydimensionality, a spatial discretization scheme must be prescribed. The knownfact that the KPZ equation can be obtained as a result of a Hopf--Coletransformation applied to a diffusion equation (with \emph{multiplicative}noise) is shown here to strongly restrict the arbitrariness in the choice ofspatial discretization schemes. On one hand, the discretization prescriptionsfor the Laplacian and the nonlinear (KPZ) term cannot be independently chosen.On the other hand, since the discretization is an operation performed on\emph{space} and the Hopf--Cole transformation is \emph{local} both in spaceand time, the former should be the same regardless of the field to which it isapplied. It is shown that whereas some discretization schemes pass bothconsistency tests, known examples in the literature do not. The requirement ofconsistency for the discretization of Lyapunov functionals is argued to be anatural and safe starting point in choosing spatial discretization schemes. Wealso analyze the relation between real-space and pseudo-spectral discreterepresentations. In addition we discuss the relevance of the Galileaninvariance violation in these consistent discretization schemes, and thealleged conflict of standard discretization with the fluctuation--dissipationtheorem, peculiar of 1D.

Discretization-related issues in the KPZ equation: Consistency, Galilean-invariance violation, and fluctuation--dissipation relation