We present an analysis of transonic solutions of the steady state 1-dimensional unipolar hydrodynamic model for semiconductors in the isoentropic case. The approach is based on construction of the orbits of the system in the electron density-electric field phase plane and on representation of discontinuous solutions of the hydrodynamic boundary value problem by a union of trajectory pieces. These pieces are related by shocks obeying jump and entropy conditions. A continuation argument in the length of the semiconductor device under consideration is applied to construct a continuum of sub- and transonic solutions, which contains at least one solution for every positive length. We also present numerical results illustrating the various possible solution profiles. For this we use a regularization of the problem, adding artificial diffusion to obtain singularly perturbed problems which are then solved numerically using continuation in the regularization parameter.
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Ascher et al. (1991) studied this question.