dimensions, the KPZ roughness exponent α decreases, implying a loss of regularity. Hence the error terms of a local numerical method (as e.g. a finite differences scheme), which are proportional to some higher-order derivative of the field, are not controlled. As a consequence, a global method such as a pseudo-spectral scheme [14–16, 36] is more adequate. Nevertheless, previous experiences found in the literature showed that finite differences schemes are still able to capture the universal features of KPZ evolution. This, together with our previous considerations, is our motivation for the present work. In Sec. II we show that the relationship established by the Hopf–Cole transformation— between the KPZ equation and a diffusion equation with multiplicative noise [2]—poses constraints on the discretization procedure. We verify the consistency of the standard (nearest-neighbor) discretization scheme and find the form of the corresponding KPZ term for a general real-space discrete Laplacian; we also present some comments regarding the mapping of KPZ into the directed polymer problem [2]. In Sec. III we analyze the problem from the perspective of the Lyapunov functional, show in what sense known prescriptions for the KPZ term from the literature fail the test, and find the corresponding consistent prescriptions. Moreover, we propose a consistent real-space discretization scheme whose accuracy is far higher than that of schemes of similar complexity in the literature. In Sec. IV we discuss the relation with the pseudo-spectral method. In Sec. V, we show that a consistent discretization scheme does not (essentially) violate the fluctuation–dissipation relation, peculiar of 1D, and discuss the role of the Galilean invariance for the discrete rep- resentations of the KPZ equation, showing that such invariance seems not to be a necessary element to define the KPZ universality class. In Sec. VI we discuss a recently introduced variational approach for the KPZ equation [26], and show that it offers a natural framework for its consistent discretization. In Section VII we present some numerical results regarding critical exponents and the violation of Galilean invariance. Section VIII contains the con- clusions and final discussions. It is worth here commenting that some preliminary results were presented in [27]. 4