The Relaxation of the Hydrodynamic Model for Semiconductors to the Drift–Diffusion Equations

We establish the convergence and consistency of approximate solutions derived by the modified Godunov scheme for the initial-boundary value problem to a simplified one-dimensional hydrodynamic model for semiconductors using the compensated compactness method. The traces of weak solutions are introduced and then the weak solutions are proved to satisfy the natural boundary conditions. The zero relaxation limit of the hydrodynamic model to the drift–diffusion model is proved when the momentum relaxation time tends to zero.