  The additivity principle allows a calculation of current fluctuations and
associated density profiles in large diffusive systems. In order to test its
validity in the weakly asymmetric exclusion process with open boundaries, we
use a numerical approach based on the density matrix renormalisation. With this
technique, we determine the cumulant generating function of the current and the
density profile corresponding to atypical currents in finite systems. We find
that these converge to those predicted by the additivity principle. No evidence
for dynamical phase transitions is found.
