Uniform-in-time Bounds for approximate Solutions of the drift-diffusion System

Marianne Bessemoulin-Chatard (1) , Claire Chainais-Hillairet (2, 3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 RAPSODI - Reliable numerical approximations of dissipative systems

In this paper, we consider a numerical approximation of the Van Roosbroeck's drift– diffusion system given by a backward Euler in time and finite volume in space discretization, with Scharfetter-Gummel fluxes. We first propose a proof of existence of a solution to the scheme which does not require any assumption on the time step. The result relies on the application of a topological degree argument which is based on the positivity and on uniform-in-time upper bounds of the approximate densities. Secondly, we establish uniform-in-time lower bounds satisfied by the approximate densities. These uniform-in-time upper and lower bounds ensure the exponential decay of the scheme towards the thermal equilibrium as shown in [3].

Domaines: Analyse numérique [math.NA]

Marianne Bessemoulin-Chatard, Claire Chainais-Hillairet. Uniform-in-time Bounds for approximate Solutions of the drift-diffusion System. Numerische Mathematik, 2019, 141 (4), pp.881-916. ⟨10.1007/s00211-018-01019-1⟩. ⟨hal-01659418v2⟩