ABSTRACT

If we set the maximum voltage to start the simulation with to be one per-cent of V maxthen we get from the previous relations and a little algebra that we would have to augment the resolution by a factor of I 0000. meaning a grid of I .6 million meshes for a 1-D simulation. It is clear that the use of an irregular mesh imposes itself at this point. For easy extension to 2-D and 3-D (which description is difl.ered to another article). we have opted for a subgrid patch which extends from the emitter to the virtual surface. ln order to be able to get to a very fine resolution close to the emitter. we set the mesh spacing so as to obtain a uniform charge density in the patch at steady-state (assuming Child-Langmuir flow). At each time step. the potential is first obtained on the main regular grid and the boundary values of the patch are interpolated from the main grid solution. The field is then solved inside the patch (note that charge density deposition is performed inside both the main grid and the patch). Using 200 cells in the patch allows for an initial applied voltage to be less than one per-cent of Vmax and we can check on Fig.4.b) that the voltage history profile obtained from the simulation is now undistinguishable from the infinitesimal Lampei-Tidenback solution from the start. This has been obtained without deterioring the current profile. as can been checked in Fig.2,c). wh1ch is almost identical to the one of Fig.2.b). The peak. however. that we may have assumed to be caused by the voltage starting at a non-zero value is still present. Our assumption was at this point that the peak is due to an underresolved front of the particle distribution.