AMaSiS 2018 Workshop: Abstracts

Analysis of a finite volume scheme discretizing drift-diffusion systems

Marianne Bessemoulin-Chatard ${}^{\text{(1)}}$ , and Claire Chainais-Hillairet ${}^{\text{(2)}}$

(1) Université Nantes, CNRS, UMR 6629 – Laboratoire Jean Leray

(2) Université Lille, CNRS, UMR 8524, Inria – Laboratoire Paul Painlevé

We are interested in the analysis of a numerical scheme discretizing drift-diffusion systems for semiconductors. The considered scheme [1] is finite volume in space, and the numerical fluxes are a generalization of the classical Scharfetter–Gummel scheme, which allows to consider both linear or nonlinear pressure laws. Using a discrete entropy method, we establish the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity. We obtain an exponential decay rate by controlling the discrete relative entropy with the entropy production [2]. This result is proved under assumptions of existence and uniform-in-time upper and lower bounds for numerical solutions, which will be obtained in the linear case thanks to an adaptation to the discrete framework of the Moser’s iteration technique [3].

Acknowledgments: The author are supported by the Inria team RAPSODI, the LabEx CEMPI (ANR-11-LABX-0007-01), the Centre Henri Lebesgue (ANR-11-LABX-0020-01) and the MoHyCon project (ANR-17-CE40-0027-01).

References

1 M. Bessemoulin-Chatard, A finite volume Scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme, Numer. Math 121 (2012), 637–670.
2 M. Bessemoulin-Chatard, and C. Chainais-Hillairet, Exponential decay of a finite volume Scheme to the thermal equilibrium for drift-diffusion systems, J. Numer. Math. 3 (2017), 147–168.
3 M. Bessemoulin-Chatard, and C. Chainais-Hillairet, Uniform-in-time bounds for approximate solutions of the drift-diffusion system, Preprint hal-01659418.