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

H. S. WIO, J. A. REVELLI; R. R. DEZA; C. ESCUDERO; M. S. DE LA LAMA

PHYSICAL REVIEW E - STATISTICAL PHYSICS, PLASMAS, FLUIDS AND RELATED INTERDISCIPLINARY TOPICS

The American Physical Society

Año: 2010 vol. 81 p. 66706 - 66717

1063-651X

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 obtainedas a result of a Hopf-Cole transformation applied to a diffusion equation
with multiplicative noise is shownhere to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, thediscretization 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 transformationis 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 theliterature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to bea natural and safe starting point in choosing spatial discretization schemes. We also analyze the relationbetween real-space and pseudospectral discrete representations. In addition we discuss the relevance of theGalilean-invariance violation in these consistent discretization schemes and the alleged conflict of standarddiscretization with the fluctuation-dissipation theorem, peculiar of one dimension.