Convergence of a finite-volume scheme for the drift-diffusion equations in 1D

We construct an approximate solution to one-dimensional nonlineardrift–diffusion equations by a finite-volume scheme. Theconvergence of the scheme is proved, which yields the globalexistence of solutions to the equations. This result is valideven in the presence of vacuum state: the densities of freecarriers of charge vanish for some times and positions, whichimplies that the continuity equations are degenerate parabolicequations. Finally, numerical simulations are performed by thefinite-volume scheme.