  Several different approaches are proposed for solving fully implicit
discretizations of a simplified Boltzmann-Poisson system with a linear
relaxation-type collision kernel. This system models the evolution of free
electrons in semiconductor devices under a low-density assumption. At each
implicit time step, the discretized system is formulated as a fixed-point
problem, which can then be solved with a variety of methods. A key algorithmic
component in all the approaches considered here is a recently developed
sweeping algorithm for Vlasov-Poisson systems. A synthetic acceleration scheme
has been implemented to accelerate the convergence of iterative solvers by
using the solution to a drift-diffusion equation as a preconditioner. The
performance of four iterative solvers and their accelerated variants has been
compared on problems modeling semiconductor devices with various electron
mean-free-path.
