On the square-root approximation finite volume scheme for nonlinear drift-diffusion equations

TL;DR

This paper introduces and analyzes a finite volume scheme for nonlinear convection-diffusion equations, ensuring stability, thermodynamic consistency, and convergence, supported by numerical simulations.

Contribution

It proposes a novel finite volume scheme for nonlinear drift-diffusion equations with proven stability, thermodynamic consistency, and convergence.

Findings

Unique discrete solution exists.

Solution bounds are preserved.

Scheme dissipates free energy as expected.

Abstract

We study a finite volume scheme for the approximation of the solution to convection diffusion equations with nonlinear convection and Robin boundary conditions. The scheme builds on the interpretation of such a continuous equation as the hydrodynamic limit of some simple exclusion jump process. We show that the scheme admits a unique discrete solution, that the natural bounds on the solution are preserved, and that it encodes the second principle of thermodynamics in the sense that some free energy is dissipated along time. The convergence of the scheme is then rigorously established thanks to compactness arguments. Numerical simulations are finally provided, highlighting the overall good behavior of the scheme.