A Maximum Principle for Drift-Diffusion Equations and the Scharfetter-Gummel Discretization

The solution of the drift-diffusion equation does not satisfy a maximum principle in general. Here it is shown that a maximum principle can be established for the so called Slotboom variable, which permits statements on uniqueness, stability, and positivity. This maximum principle is preserved for the discretized system obtained by the Scharfetter-Gummel scheme.