Space-charge limited current flow: An analytical verification solution for kinetic and fluid simulations

Verification of numerical simulations is an important step in code development as it demonstrates the correctness of the code in solving the underlying physical model. Analytical solutions represent a strong tool in code verification, but due to the complexity of the fundamental equations, such solutions are often not available. This is particularly true in the case of kinetic models. Here we present a family of fully analytical solutions describing current transmission between two electrodes and which apply to both fluid, and kinetic, descriptions of the system. The solutions account for the finite initial particle injection velocity and are valid for injection currents between zero and the maximum at the space-charge limit. In addition to determining this space-charge limited current, spatial profiles of all physical quantities (such as the particle density and velocity) are also obtained. This provides a means to not only verify fluid and kinetic simulations, but also to assess the error and accuracy of the numerical simulation parameters used. The analytical solutions extend the classical Child-Langmuir law (which only applies to an initial injection velocity equal to zero), and provide new insight into space-charge limited current flow.