  The dual boson approach [Ann. Phys. 327, 1320 (2012)] provides a means to
construct a diagrammatic expansion around the extended dynamical mean-field
theory (EDMFT). In this paper, we present the numerical implementation of the
approach and apply it to the extended Hubbard model with nearest-neighbor
interaction $V$. We calculate the EDMFT phase diagram and study the effect of
diagrams beyond EDMFT on the transition to the charge-ordered phase. Including
diagrammatic corrections to the EDMFT polarization shifts the EDMFT phase
boundary to lower values of $V$. The approach interpolates between the random
phase approximation in the weak coupling limit and EDMFT for strong coupling.
Neglecting vertex corrections leads to results reminiscent of the EDMFT+$GW$
approximation. We however find significant deviations from the dual boson
results already for small values of the interaction, emphasizing the crucial
importance of fermion-boson vertex corrections.
