Large-time behavior of a family of finite volume schemes for boundary-driven convection-diffusion equations

TL;DR

This paper demonstrates that a wide class of finite volume schemes for boundary-driven convection-diffusion equations preserve exponential decay to equilibrium, ensuring accurate long-term behavior in numerical simulations.

Contribution

It proves that many finite volume schemes maintain exponential decay to steady state for various convection-diffusion models with boundary conditions.

Findings

Finite volume schemes preserve exponential decay to steady state.

Theoretical results validated on multiple numerical tests.

Applicable to models like Fokker-Planck and drift-diffusion systems.

Abstract

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, porous media equations or drift-diffusion systems for semiconductors. For all of these models, some relative entropy principle is satisfied and implies exponential decay to the stationary state. In this paper we show that in the framework of finite volume schemes on orthogonal meshes, a large class of two-point monotone fluxes preserve this exponential decay of the discrete solution to the discrete steady state of the scheme. This includes for instance upwind and centered convections or Scharfetter-Gummel discretizations. We illustrate our theoretical results on several numerical test cases.