  We recently have presented first physical predictions of a spatially hybrid
model that follows the evolution of a negative streamer discharge in full three
spatial dimensions; our spatially hybrid model couples a particle model in the
high field region ahead of the streamer with a fluid model in the streamer
interior where electron densities are high and fields are low. Therefore the
model is computationally efficient, while it also follows the dynamics of
single electrons including their possible run-away. Here we describe the
technical details of our computations, and present the next step in a
systematic development of the simulation code. First, new sets of transport
coefficients and reaction rates are obtained from particle swarm simulations in
air, nitrogen, oxygen and argon. These coefficients are implemented in an
extended fluid model to make the fluid approximation as consistent as possible
with the particle model, and to avoid discontinuities at the interface between
fluid and particle regions. Then two splitting methods are introduced and
compared for the location and motion of the fluid-particle-interface in three
spatial dimensions. Finally, we present first results of the 3D spatially
hybrid model for a negative streamer in air.
