  We investigate the acceleration of electrons via their interaction with
electrostatic waves, driven by the relativistic Buneman instability, in a
system dominated by counter-propagating proton beams. We observe the growth of
these waves and their subsequent saturation via electron trapping for a range
of proton beam velocities, from 0.15c to 0.9c. We can report a reduced
stability of the electrostatic wave (ESW) with increasing non-relativistic beam
velocities and an improved wave stability for increasing relativistic beam
velocities, both in accordance with previous findings. At the highest beam
speeds, we find the system to be stable again for a period of ~160 plasma
periods. Furthermore we observe a, to our knowledge, previously unreported
secondary electron acceleration mechanism at low beam speeds. We believe that
it is the result of parametric couplings to produce high phase velocity ESW's
which then trap electrons, accelerating them to higher energies. This allows
electrons in our simulation study to achieve the injection energy required for
Fermi acceleration, for beam speeds as low as 0.15c in unmagnetised plasma.
