Abstract: This talk describes solution of the Child-Langmuir problem of the continuum flow of electrons from cathode to anode across a channel by two methods: The first is direct numerical simulation. The second is a new analytic method. When put into characteristic variables - i.e., variables that follow a particle path - the equations admit a simple analytic, but implicit, solution for any choice of boundary data for velocity **v**, electric field E and current J at the cathode. For steady solutions, this approach clarifies the origins of the maximal current J_{max} for a given potential jump across the channel, as derived by Child and Langmuir for incoming velocity **v**=0 and by Jaffe for v>0. Moreover, it leads to an approximate form for the time-dependent solution when J>J_{max}. The results are compared to those from direct numerical simulation. |