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Large time behavior of nonlinear finite volume schemes for convection-diffusion equations

Clément Cancès 1, 2 Claire Chainais-Hillairet 2, 1 Maxime Herda 1, 2 Stella Krell 3, 4
1 RAPSODI - Reliable numerical approximations of dissipative systems
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe
4 COFFEE - COmplex Flows For Energy and Environment
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR7351
Abstract : In this contribution we analyze the large time behavior of a family of nonlinear finite volume schemes for anisotropic convection-diffusion equations set in a bounded bidimensional domain and endowed with either Dirichlet and / or no-flux boundary conditions. We show that solutions to the two-point flux approximation (TPFA) and discrete duality finite volume (DDFV) schemes under consideration converge exponentially fast toward their steady state. The analysis relies on discrete entropy estimates and discrete functional inequalities. As a biproduct of our analysis, we establish new discrete Poincaré-Wirtinger, Beckner and logarithmic Sobolev inequalities. Our theoretical results are illustrated by numerical simulations.
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Submitted on : Tuesday, June 9, 2020 - 10:55:17 AM
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Clément Cancès, Claire Chainais-Hillairet, Maxime Herda, Stella Krell. Large time behavior of nonlinear finite volume schemes for convection-diffusion equations. SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, In press. ⟨hal-02360155v2⟩



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