On the effects of boundary conditions at the cathode on the equations of steady space-charge flow

The steady, zero-temperature, space-charge flow problem has a unique solution if formulated in the differential equation form. However, if the differential equations are approximated by a difference equation formulation, it is shown that the solution is no longer unique. The effect is demonstrated by the example of the Langmuir planar diode. It is proved theoretically for the planar diode, and demonstrated numerically for a two-dimensional example, that the ambiguity in the perveance of a gun due to this effect is of order
h/d
; here
h
is the mesh size of the difference mesh, and
d
is the mean length of the particle trajectories.