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 Schrödinger–Poisson 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.