A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure

TL;DR

This paper introduces a finite volume scheme for nonlinear parabolic equations with boundary conditions that preserve steady-states and entropy structures, ensuring accurate long-term behavior in simulations of physical models.

Contribution

The paper develops a novel finite volume discretization that maintains steady-states and entropy properties for boundary-driven convection-diffusion equations, with proven stability and convergence.

Findings

Scheme preserves steady-states and entropy functionals.

Proven exponential convergence to equilibrium.

Numerical results confirm accuracy and efficiency.

Abstract

We propose a finite volume scheme for a class of nonlinear parabolic equations endowed with non-homogeneous Dirichlet boundary conditions and which admit relative en-tropy functionals. For this kind of models including porous media equations, Fokker-Planck equations for plasma physics or dumbbell models for polymer flows, it has been proved that the transient solution converges to a steady-state when time goes to infinity. The present scheme is built from a discretization of the steady equation and preserves steady-states and natural Lyapunov functionals which provide a satisfying long-time behavior. After proving well-posedness, stability, exponential return to equilibrium and convergence, we present several numerical results which confirm the accuracy and underline the efficiency to preserve large-time asymptotic.