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

TL;DR

This paper studies the long-term behavior of nonlinear finite volume schemes for anisotropic convection-diffusion equations, showing exponential convergence to steady states using discrete entropy and functional inequalities.

Contribution

It provides the first analysis of large time behavior for these schemes, introducing new discrete inequalities and demonstrating exponential convergence.

Findings

Solutions converge exponentially fast to steady state.

New discrete Poincaré-Wirtinger, Beckner, and logarithmic Sobolev inequalities are established.

Numerical simulations confirm theoretical results.

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 Poincar\'e-Wirtinger, Beckner and logarithmic Sobolev inequalities. Our theoretical results are illustrated by numerical simulations.