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

TL;DR

This paper investigates the constraints on spatial discretization schemes for the KPZ equation, emphasizing the importance of consistency, Galilean invariance, and fluctuation-dissipation relations, and highlights issues with existing schemes.

Contribution

It establishes the strong restrictions on discretization schemes derived from the Hopf--Cole transformation and proposes consistency of Lyapunov functionals as a guiding principle.

Findings

Some discretization schemes satisfy consistency tests

Existing schemes in literature often violate these consistency conditions

Discussion on Galilean invariance violation and fluctuation-dissipation relation in discretizations

Abstract

In order to perform numerical simulations of the 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 \emph{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 \emph{space} and the Hopf--Cole transformation is \emph{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 pseudo-spectral 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 1D.