Convergence of a finite volume scheme for a system of interacting species with cross-diffusion
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Numerische Mathematik
Convergence of a finite volume scheme for a system of interacting species with cross-diffusion José A. Carrillo1 · Francis Filbet2 · Markus Schmidtchen3 Received: 6 October 2019 / Revised: 26 March 2020 © The Author(s) 2020
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. Mathematics Subject Classification Primary 74S10 · 65M12 · 92C15; Secondary 45K05 · 92D25 · 47N60
1 Introduction In this paper we develop and analyse a numerical scheme for the following non-local interaction system with cross-diffusion and self-diffusion
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José A. Carrillo [email protected] Francis Filbet [email protected] Markus Schmidtchen [email protected]
1
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
2
Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France
3
Laboratoire Jacques-Louis Lions, Sorbonne Université, 4 place Jussieu, 75005 Paris, France
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J. A. Carrillo et al.
∂ ∂ρ 2 ρ , (W11 ρ + W12 η + ν(ρ + η)) + ∂x 2 ∂x ∂η ∂ ∂ ∂η2 = η , (W22 η + W21 ρ + ν(ρ + η)) + ∂t ∂x ∂x 2 ∂x ∂ ∂ρ = ∂t ∂x
(1)
governing the evolution of two species ρ and η on an interval (a, b) ⊂ R for t ∈ [0, T ). The system is equipped with nonnegative initial data ρ 0 , η0 ∈ L 1+ (a, b) ∩ L ∞ + (a, b). We denote by m 1 the mass of ρ0 and by m 2 the mass of η0 , respectively, m1 = a
b
ρ0 (x) dx, and m 2 =
b
η0 (x) dx.
a
On the boundary x = a and b, we prescribe no-flux boundary conditions ∂ (W11 ρ + W12 η + ν(ρ + η) + ρ) = 0, ∂x ∂ η (W22 η + W21 ρ + ν(ρ + η) + η) = 0, ∂x
ρ
such that the total mass of each species is conserved with respect to time t ≥ 0. While the self-interaction potentials W11 , W22 ∈ Cb2 (R) model the interactions among individuals of the same species (also referred to as intraspecific interactions), the crossinteraction potentials W12 , W21 ∈ Cb2 (R) encode the interactions between individuals belonging to different species, i.e. interspecific interactions. Here Cb2 (R) denotes the set of twice continuously differentiable functions on R with bounded derivatives. Notice that the convolutions Wi j ψ, with ψ a density function defined on [a, b], are defined by extending the density ψ by zero outside the interval [a, b]. The two positive parameters , ν > 0 determine the strengths of the self-diffusion and the cross-diffusion of both species, respectively. Nonlinear diffusion, be it self
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