A convergent entropy-dissipating BDF2 finite-volume scheme for a population cross-diffusion system

TL;DR

This paper develops a second-order BDF2 finite-volume scheme for a nonlinear population cross-diffusion system, ensuring entropy dissipation, mass conservation, and convergence, with proofs and numerical validation.

Contribution

It extends the entropy inequality to the system case and proves convergence and stability of the scheme for population dynamics models.

Findings

The scheme preserves Rao entropy and mass.

Existence and uniqueness of discrete solutions are established.

Numerical experiments confirm theoretical results.

Abstract

A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates. The novelty is the extension of this inequality to the system case. Some numerical experiments in one and two space dimensions underline the theoretical results.