AMaSiS 2018 Workshop: Abstracts

Relative entropy properties of monotone schemes for boundary-driven convection-diffusions

Claire Chainais-Hillairet${}^{\mathbf{\text{(1)}}}$ and Maxime Herda${}^{\mathbf{\text{(2)}}}$

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

(2) Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions

We are interested in the large-time behavior of solutions to finite volume discretizations of convection-diffusion equations or systems endowed with non-homogeneous Dirichlet and Neumann type boundary conditions.

Our results concern various linear and nonlinear models such as Fokker-Planck equations, the porous medium equation or drift-diffusion system for semiconductors. For all these models, some relative entropy principle is satisfied and implies exponential decay to the stationary state.

In this talk we will provide some elements of proof to show that in the framework of finite volume discretization on orthogonal meshes, a large class of two-point monotone flux preserve this exponential decay of the discrete solution to the discrete stationary solution of the scheme. This includes for instance upwind and centered convections or Scharfetter-Gummel discretizations.

The key tools are the adaptation to the discrete level of the entropy method proposed by Bodineau, Lebowitz, Mouhot and Villani in [1] and a rewritting of the flux using the steady state in order to apply the strategy of proof developed by Filbet and Herda in [2].

We will illustrate numerically the validity of our results for various test cases and schemes.

Acknowledgments: The work of Maxime Herda is supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098, LabEx SMP )