In this talk we first review the critical threshold phenomena for Euler-Poisson equations, which arise in the semiclassical approximation of Schrodinger-Poisson equations and plasma dynamics. We then present a phased space-based level set method for the computation of multi-valued velocity and electric fields of one-dimensional Euler-Poisson equations. This method uses an implicit Eulerian formulation in an extended space, which incorporates both velocity and electric fields into the configuration space. Multi-valued velocity and electric fields are captured through common zeros of two level set functions, which solve a linear homogeneous transport equation in the field space. The superposition principle for multi-valued solutions is established.