We propose a direct solver to the non-stationary Boltzmann-Poisson system for simulating the electron transport in two-dimensional GaAs devices. The GaAs conduction band is approximated by a two-valley model. All of the important scattering mechanisms are taken into account. Our numerical scheme consists of the combination of the multigroup approach to deal with the dependence of the electron distribution functions on the three-dimensional electron wave vectors and a high-order WENO reconstruction procedure for treating their spatial dependences. The electric field is determined self-consistently from the Poisson equation. Numerical results are presented for a GaAs-MESFET. We display electron distribution functions as well as several macroscopic quantities and compare them to those of Monte Carlo simulations. In addition, we study the influence of the used discretization on the obtained results.