A field-space-based level set method for computing multi-valued solutions to 1D Euler–Poisson equations

We present a field-space-based level set method for computing multi-valued solutions to one-dimensional Euler–Poisson equations. The system of these equations has many applications, and in particular arises in semiclassical approximations of the Schrödinger–Poisson equation. The proposed approach involves an implicit Eulerian formulation in an augmented space – called field space, which incorporates both velocity and electric fields into the configuration. Both velocity and electric fields are captured through common zeros of two level set functions, which are governed by a field transport equation. Simultaneously we obtain a weighted density f by solving again the field transport equation but with initial density as starting data. The averaged density is then resolved by the integration of the obtained f against the Dirac delta-function of two level set functions in the field space. Moreover, we prove that such obtained averaged density is simply a linear superposition of all multi-valued densities; and the averaged field quantities are weighted superposition of corresponding multi-valued ones. Computational results are presented and compared with some exact solutions which demonstrate the effectiveness of the proposed method.