  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 non linear regime. We also investigate extensions of the
Schr\"odinger-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.
