A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter-Gummel scheme

TL;DR

This paper introduces a finite volume scheme extending the Scharfetter-Gummel method to nonlinear diffusion equations, effectively handling degenerate cases and preserving steady-states, with proven convergence and demonstrated efficiency in physical models.

Contribution

It develops a novel finite volume scheme for nonlinear convection-diffusion equations, ensuring stability, convergence, and steady-state preservation, applicable to physical systems like semiconductors and porous media.

Findings

Scheme converges in nondegenerate cases.

Scheme preserves steady-states.

Numerical simulations confirm efficiency and long-time accuracy.

Abstract

We propose a finite volume scheme for convection-diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter-Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.