Exponential decay of a finite volume scheme to the thermal equilibrium for drift--diffusion systems

TL;DR

This paper analyzes a finite volume numerical scheme for drift-diffusion systems in semiconductors, demonstrating exponential decay to thermal equilibrium over time with convergence rates supported by entropy methods.

Contribution

It introduces a generalized Scharfetter--Gummel scheme for nonlinear pressure laws and proves exponential decay to equilibrium using discrete entropy methods.

Findings

Proves exponential decay of the scheme to thermal equilibrium

Establishes convergence rates based on entropy production

Provides numerical illustrations confirming theoretical results

Abstract

In this paper, we study the large--time behavior of a numerical scheme discretizing drift-- diffusion systems for semiconductors. The numerical method is finite volume in space, implicit in time, and the numerical fluxes are a generalization of the classical Scharfetter-- Gummel scheme which allows to consider both linear or nonlinear pressure laws. We study the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity, and obtain a decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniform-in-time $L $\infty$$ estimates for numerical solutions, which are then discussed. We conclude by presenting some numerical illustrations of the stated results.