  We consider the full-duplex two-way relay channel with direct link between
two users and propose two coding schemes: a partial decode-forward scheme, and
a combined decode-forward and compute-forward scheme. Both schemes use
rate-splitting and superposition coding at each user and generate codewords for
each node independently. When applied to the Gaussian channel, partial
decode-forward can strictly increase the rate region over decode-forward, which
is opposite to the one-way relay channel. The combined scheme uses
superposition coding of both Gaussian and lattice codes to allow the relay to
decode the Gaussian parts and compute the lattice parts. This scheme can also
achieve new rates and outperform both decode-forward and compute-forward
separately. These schemes are steps towards understanding the optimal coding.
