We explore the use of field solvers as approximations of classical Vlasov-Poisson systems. This correspondence is investigated in both electrostatic and gravitational contexts. We demonstrate the ability of field solvers to be excellent approximations of problems with cold initial condition into the nonlinear regime. We also investigate extensions of the Schrodinger-Poisson system that employ multiple stacked cold streams, and the von Neumann-Poisson equation as methods that can successfully reproduce the classical evolution of warm initial conditions. We then discuss how appropriate simulation parameters need to be chosen to avoid interference terms, aliasing, and wave behavior in the field solver solutions. We present a series of criteria clarifying how parameters need to be chosen in order to effectively approximate classical solutions.