Numerical solutions are given for an electrostatic model of an auroral arc. This model, as first proposed, was a phenomenological model based on a variety of auroral and satellite observations. The numerical calculations reported here provide theoretical support for the model by showing that it is consistent with well-known physical laws. Two-dimensional solutions to the nonlinearly coupled Poisson-Vlasov system are exhibited for an electrostatic shock in which the equipotentials assume the form of a V. The shock is supported by a spatially localized electric current and is capable of accelerating electrons to energies large in comparison to energies of particles entering the shock system. The electrons are assumed to be strongly magnetized, while adiabatic particle orbit theory is used to calculate the ion space charge density. Attempts to calculate shock structure from a model consisting of only cold interpenetrating ion and electron beams were unsuccessful. The addition of thermal populations of particles to the beam plasmas was found to be necessary in order to obtain solutions.