content_image/366214.jpg content_image/366212.jpg content_image/366213.jpg content_image/366211.jpg content_image/366210.jpg content_image/366209.jpg
## A. Galilean Invariance 

Galilean invariance means that the KPZ equation is invariant under the transformation 

<img src='content_image/366214.jpg'>

where v is an arbitrary constant vector field and F is the external (constant) driving force. Using the classical discretization 

for the (complete) KPZ equation, we find 

<img src='content_image/366212.jpg'>

<img src='content_image/366213.jpg'>

One can immediately check that this equation is invariant under the discrete Galilean trans- formation 

<img src='content_image/366211.jpg'>

However, Eq. (43) has been criticized for its instability properties, at least when the spatial discretization is not fine enough [9]. If we use the alternative discretization 

<img src='content_image/366210.jpg'>

we find that this equation is not invariant under the discrete Galilean transformation. In fact, the transformation h → h + vja yields an excess term which is compatible with the gradient discretization in Eq. (42); however, this discretization does not allow to recover the quadratic term in Eq. (45), indicating that this finite differences scheme does not fulfill Galilean invariance. The Hopf–Cole transformed equation 

<img src='content_image/366209.jpg'>

