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

2019/11/12 by Clément Cancès, Cancès, Clément, Claire Chainais-Hillairet +5 · 1 citation
Engineering · Mathematics · #35K51 #35Q84 #39B62 #65M08 #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1911.04962

openalex publication_date 2019/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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