Study of an entropy dissipating finite volume scheme for a nonlocal cross-diffusion system

Maxime Herda (1, 2) , Antoine Zurek (3)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne

In this paper we analyse a finite volume scheme for a nonlocal version of the Shigesada-Kawazaki-Teramoto (SKT) cross-diffusion system. We prove the existence of solutions to the scheme, derive qualitative properties of the solutions and prove its convergence. The proofs rely on a discrete entropy-dissipation inequality, discrete compactness arguments, and on the novel adaptation of the so-called duality method at the discrete level. Finally, thanks to numerical experiments, we investigate the influence of the nonlocality in the system: on convergence properties of the scheme, as an approximation of the local system and on the development of diffusive instabilities.

Mots clés: Nonlocal cross-diffusion; finite volume schemes; entropy method; convergence

Domaines: Analyse numérique [math.NA]

Maxime Herda, Antoine Zurek. Study of an entropy dissipating finite volume scheme for a nonlocal cross-diffusion system. ESAIM: Mathematical Modelling and Numerical Analysis, 2023, 57 (3), pp.1589 - 1617. ⟨10.1051/m2an/2023032⟩. ⟨hal-03714164v2⟩