  A simple upwind discretization of the highly coupled non-linear differential
equations which define the hydrodynamic model for semiconductors is given in
full detail. The hydrodynamic model is able to describe inertia effects which
play an increasing role in different fields of opto- and microelectronics. A
silicon $n^+ - n - n^+$ - structure is simulated, using the energy-balance
model and the full hydrodynamic model. Results for stationary cases are then
compared, and it is pointed out where the energy-balance model, which is
implemented in most of today's commercial semiconductor device simulators,
fails to describe accurately the electron dynamics. Additionally, a GaAs $n^+ -
n - n^+$-structure is simulated in time-domain in order to illustrate the
importance of inertia effects at high frequencies in modern submicron devices.
