  We present and analyze a new iterative solver for implicit discretizations of
a simplified Boltzmann-Poisson system. The algorithm builds on recent work that
incorporated a sweeping algorithm for the Vlasov-Poisson equations as part of
nested inner-outer iterative solvers for the Boltzmann-Poisson equations. The
new method eliminates the need for nesting and requires only one transport
sweep per iteration. It arises as a new fixed-point formulation of the
discretized system which we prove to be contractive for a given electric
potential. We also derive an accelerator to improve the convergence rate for
systems in the drift-diffusion regime. We numerically compare the efficiency of
the new solver, with and without acceleration, with a recently developed nested
iterative solver.
