AMaSiS 2021 - Abstract

Moatti, Julien

Long-time behaviour of a hybrid finite volume scheme for the drift-diffusion model with magnetic field

Coauthors: Claire Chainais-Hillairet, Maxime Herda and Simon Lemaire
Inria, Université de Lille, France

In this talk, we introduce a Hybrid Finite Volume (HFV) scheme to discretise the isothermal drift-diffusion system for semiconductors.
The HFV schemes [1] - generalisations of classical two-point finite volume schemes - are devised to handle general polygonal/polyhedral meshes, alongside with anisotropic diffusion tensors. Especially, the scheme introduced here can be used in situations where the semiconductor is immersed in a magnetic field [2].
The scheme is based on the nonlinear discretisation introduced in [3]. Its analysis relies on the preservation of a discrete entropy structure, which mimics the continuous behaviour of the system. Using these properties, we show the existence of solutions to the scheme, and ensure the positivity of the carrier densities. Moreover, we establish the convergence of the discrete solution towards a discrete thermal equilibrium as time tends to infinity.
We will give some numerical illustration of our theoretical results.

Acknowledgments: The authors are supported by the Inria team RAPSODI and the LabEx CEMPI (ANR-11-LABX-0007-01)

Refrences
[1] R. Eymard, T. Gallouët,,and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009--1043.
[2] H. Gajewski, and K. Gärtner, On the Discretization of van Roosbroeck's Equations with Magnetic Field, ZAMM - Journal of Applied Mathematics and Mechanics, 11 (1995), 247--264.
[3] C. Chainais-Hillairet, M. Herda, S. Lemaire, and J. Moatti, Long-time behaviour of hybrid finite volume schemes for advection-diffusion equations: linear and nonlinear approaches, in preparation.