CONVERGENCE OF A FINITE VOLUME SCHEME FOR A SYSTEM OF INTERACTING SPECIES WITH CROSS-DIFFUSION

José Carrillo (1) , Francis Filbet (2) , Markus Schmidtchen (1)

1 Department of Mathematics [Imperial College London]
2 IMT - Institut de Mathématiques de Toulouse UMR5219

In this work we present the convergence of a positivity preserving semi-discrete finite volume scheme for a coupled system of two non-local partial differential equations with cross-diffusion. The key to proving the convergence result is to establish positivity in order to obtain a discrete energy estimate to obtain compactness. We numerically observe the convergence to reference solutions with a first order accuracy in space. Moreover we recover segregated stationary states in spite of the regularising effect of the self-diffusion. However, if the self-diffusion or the cross-diffusion is strong enough, mixing occurs while both densities remain continuous.

Domaines: Analyse numérique [math.NA]

José Carrillo, Francis Filbet, Markus Schmidtchen. CONVERGENCE OF A FINITE VOLUME SCHEME FOR A SYSTEM OF INTERACTING SPECIES WITH CROSS-DIFFUSION. Numerische Mathematik, 2020, 145, pp.473-511. ⟨hal-01764444⟩