  We study the orbital evolution and gravitational wave (GW) emission of
supermassive black hole (SMBH) binaries formed in gas-free mergers of massive
early-type galaxies using the hybrid tree-regularized N-body code KETJU. The
evolution of the SMBHs and the surrounding galaxies is followed
self-consistently from the large-scale merger down to the final few orbits
before the black holes coalesce. Post-Newtonian corrections are included up to
PN3.5-level for the binary dynamics, and the GW calculations include the
corresponding corrections up to PN1.0-level. We analyze the significance of the
stellar environment on the evolution of the binary and the emitted GW signal
during the final GW emission dominated phase of the binary hardening and
inspiral. Our simulations are compared to semi-analytic models that have often
been used for making predictions for the stochastic GW background emitted by
SMBHs. We find that the commonly used semi-analytic parameter values produce
large differences in merger timescales and eccentricity evolution, but result
in only $\sim 10\%$ differences in the GW spectrum emitted by a single binary
at frequencies $f\gtrsim 10^{-1} \, \rm yr^{-1}$, which are accessible by
current pulsar timing arrays. These differences are in part caused by the
strong effects of the SMBH binaries on the surrounding stellar population,
which are not included in the semi-analytic models.
