In this work, the ionic motion in an aqueous electrolyte solution is studied within the framework of a fully self-consistent Langevin-Poisson solver in order to verify the accuracy of the approach. The primitive model is used to describe the individual ions as charged spheres moving in a continuum solvent. The P3M method is used to self-consistently resolve the electrostatic behavior of both the long-range forces of the collective plasma and the boundary conditions, and the short-range inter-particle interactions resulting from the Coulombic force between close ions. A small test volume representing a portion of the large aqueous electrolyte solution is simulated to calibrate the simulation tool under nonequilibrium conditions. Results of the conductivity of NaCl and KCl solutions are presented for several concentrations and the radial distribution functions in these liquids is discussed.