Computing Multi-valued Velocity and Electric Fields for 1d Euler-poisson Equations

We develop a level set method for the computation of multi-valued velocity and electric fields of one-dimensional Euler-Poisson equations. The system of these equations arises in the semiclassical approximation of SchrödingerPoisson equations and semiconductor modeling. This method uses an implicit Eulerian formulation in an extended space — called field 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. Numerical examples are presented to validate the proposed level set method.