Exponential decay of a finite volume scheme to the thermal equilibrium for drift--diffusion systems
For researchers in numerical analysis of semiconductor models, this provides a rigorous convergence guarantee for a widely used numerical scheme, though the result is incremental as it extends existing entropy-based techniques to a generalized Scharfetter-Gummel scheme.
The paper proves that a finite volume scheme for drift-diffusion semiconductor systems converges to an approximation of thermal equilibrium with an exponential decay rate, using discrete relative entropy control. Numerical illustrations confirm the theoretical result.
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.