Abstract
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.
Highlights
In this paper we develop and analyse a numerical scheme for the following non-local interaction system with cross-diffusion and self-diffusion
At the same time we would like to stress that the self-diffusion terms are relevant for the convergence analysis below. It is the interplay between the non-local interactions of both species and their individual and joint size-exclusion, modelled by the non-linear diffusion [4,5,6,10,12, 40], that leads to a large variety of behaviours including complete phase separation or mixing of both densities in both stationary configurations and travelling pulses [11,19]
One part comes from the cross-diffusion part, and the second one comes from the non-local interaction fields
Summary
Governing the evolution of two species ρ and η on an interval (a, b) ⊂ R for t ∈ [0, T ). At the same time we would like to stress that the self-diffusion terms are relevant for the convergence analysis below It is the interplay between the non-local interactions of both species and their individual and joint size-exclusion, modelled by the non-linear diffusion [4,5,6,10,12, 40], that leads to a large variety of behaviours including complete phase separation or mixing of both densities in both stationary configurations and travelling pulses [11,19]. Convergence of a finite volume scheme for a system of This includes reaction-(cross-)diffusion systems [3,7,17] and references therein, and by adding non-local interactions [2,11,19,24] and references therein. In choosing 0 < α < we obtain d dt b ρ log ρ + η log η a dx + ν b a
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