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

We construct an approximate solution to one-dimensional nonlinear drift-diffusion equations by a finite-volume scheme. The convergence of the scheme is proved, which yields the global existence of solutions to the equations. This result is valid even in the presence of vacuum state: the densities of free carriers of charge vanish for some times and positions, which implies that the continuity equations are degenerate parabolic equations. Finally, numerical simulations are performed by the finite-volume scheme.