  A general framework for the field-theoretic thermodynamic uncertainty
relation was recently proposed and illustrated with the $(1+1)$ dimensional
Kardar-Parisi-Zhang equation. In the present paper, the analytical results
obtained there in the weak coupling limit are tested via a direct numerical
simulation of the KPZ equation with good agreement. The accuracy of the
numerical results varies with the respective choice of discretization of the
KPZ non-linearity. Whereas the numerical simulations strongly support the
analytical predictions, an inherent limitation to the accuracy of the
approximation to the total entropy production is found. In an analytical
treatment of a generalized discretization of the KPZ non-linearity, the origin
of this limitation is explained and shown to be an intrinsic property of the
employed discretization scheme.
