Field-limited emission is an approach for using Gauss's Law to to satisfy the space charge limit for emitting current in Particle-in-Cell simulations. A thorough analysis is done of the assumptions made in prior implementations, and corrections are introduced for cylindrical geometry, non-zero injection velocity, and non-zero surface field. Particular care is taken to determine special conditions for the outermost node, where we find that forcing a balance of Gauss's Law would be incorrect. The scheme is then applied to determine a scaling law for the Child-Langmuir limit in an axisymmetric planar diode, with finite initial velocity. We find that the new scheme reproduces prior results, in significantly less computation time, due to no longer needing to overinject and to rapid convergence of the surface field using our new algorithmic optimization wrapper to seek the local limiting current along an emitter.
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