  Using a combination of Eulerian and Lagrangian variables we obtain some exact
results and good approximation schemes for the time evolution of the electron
flow from a no-current state to a final stationary current state in a planar
one-dimensional diode. The electrons can be injected externally or generated by
the cathode via field emission governed by a current-field law. The case of
equipotential electrodes and fixed injection is studied along with a positive
anode potential. When the current is fixed externally the approach to the
stationary state goes without oscillations if the initial electron velocity is
high enough and the anode can absorb the injected flow. Otherwise the
accumulated space charge creates a potential barrier which reflects the flow
and leads to its oscillations, but our method of analysis is invalid in such
conditions. In the field emission case the flow goes to its stationary state
through a train of decaying oscillations whose period is of the order of the
electron transit time, in agreement with earlier studies based on perturbation
techniques. Our approximate method does not permit very high cathode
emissivity, though the method works when the stationary current density is only
about 10% smaller than the Child-Langmuir limit.
